A TEXT BOOK OF HEAT
FOR JUNIOR STUDENTS
(INCLUDING KINETIC THEORY OF GASES,
THERMODYNAMICS AND RADIATION)
BY
M. N. SAHA, ftSc, F.R.S.,
Late Palit Professor of Physics, Calcutta Universtty
AND
B. N, SRIVASTAVA, D.Sc^ F.N.I.,
Professor of General Physics, Indian Association for the
Cultivation of Science, Calcutta
(TWELFTH EDITION)
SCIENCE BOOK AGENCY
I*. I'i'ili, Lake Terrace, CaIcutta29
First Edition
Second Impression
Sf.co.nd Edition
1939
Third Edition
1943
Fourth Edition
1945
Fifth Edition
1949
Sixth Edition
1951
Seventh Edition
1953
Eighth Edition
1954
ON
1'iuntr Edition
ELEVENTH !'n
Twelfth
ill Rights resen
Publishku by Sm, Roma Saita, ] : ihern Avenue,
Calcutta :
Printed by Modern India Press
7, Raja Sudodh Mullick Square, Caicutta 13
Price Rupees Nine only
PREFACE TO THE FIRST EDITION
i Text. Book of Heat for Junior Students has been written with
iv to supplying the needs of the students of the pass course
ir the Bachelor's degree. It has grown out of the lectures
which i he senior author has been giving to die B.Sc, pass class of
Ulahabad University for several years. The plan of
closely follows that of the larger Text Book which is intended for
B.Sc. honours and M.Sc. students. Separate chapters have been
i i voted to Kinetic Theory, Liquefaction of Gases and Heat. Engines,
principles of Thermodynamics and their applications have been
Created at considerable length. Throughout the book the methods of
calculus have been freely employed. The supplementary chapter on
orology has been kindly written by Dr. A. K. Das of the Indian
Meteorological Sendee and Mr. B. N\ Srivasuava. Meteorology is a
growing science and is extremely useful to die public at large. It is
not at present included in the curriculum of any Indian University
il Agra where it forms a special course for the M«Sc. degree) ,
hut this seems to be a cardinal omission. It is hoped that in future
it will form a regular subject of study by degree students,
\& this is the first edition of the book, it is feared that there
may he several omissions and inaccuracies. The authors will be
grateful if these arc brought to their notice.
Allahabad : M. N, S.
maty, 1933. B* N. S.
PREFACE TO THE ELEVENTH EDITION
Since the last edition many Indian Universities have Introduced
the new threeyear Degree Course while some others are still continu
ing the old twoyear course. The book has therefore been thoroughly
revised to cover the syllabus of the new threeyear course of most
Indian universities. As this required only addition of some matter
previously found in the Intermediate Syllabus and as the old subject
matter of the book has been almost wholly retained, it is confidently
hoped that the book will prove equally useful both for the new three
course and the old twoveai course.
Calcutta :
Juk t 1962.
B. N. S.
PREFACE TO THE TWELFTH EDITION
Several suggestions for the improvement of the book, kindly
i hi by teachers using the book, have been incorporated in this
edition. The .standard questions, arranged chapterwise and given at
the end o! the book have been brought uptndate.
I. una :
luajy, J 967. B, N, &
ACKNOWLEDGMENTS
CONTENTS
We have much pleasure in expressing our indebtedness to the
following authors, publishers and societies for allowing us to reproduce
diagrams which appeared in the works mentioned below : —
MullerPouillets Lehrbuch der Physik,
Heading, Temperat itrmessung,
Nernst, Grundlagen des neuen Warmesatzes.
Jellinek, Lehrbuch der physiftalischen Chemie.
Ewing, The Steam Engine and Other Heat Engines.
Watson, Practical Physics.
Burgess and Le Chatclicr, The Measurement of High Temperatures.
Proceedings of the Physical Society of London,
Physical Review.
Journal of the Optical Society of America, Vol. 10.
Proceedings of the Royal Society, London, Chap, II, Figs, 4, 5 ;
Chap. VII, Fisr. 10.
Philosophical lions of the Royal Society, Chap. II, Fi
Char, vn, Fig, io.
Zeiti , V, Fig,
: ; , Chap, i I Fig. 16.
Atmai >' ; Chap, VII, Fig. 3.
[buck d< Chap, IV. Fig. 7; Chap. VI, Fig. 10;
Chap. XL Fig.
Handbuch der Experi physik. Chap. V, Fig. 11; Chap. VI,
Pigi tap. Vni, Fig, ' hap. XT, Yi^. 11, 18, 19, 20.
Ezer I (hods of Measuring Temperature. Chap. I, Figs.
5, 6, 7, 9; Chap. XT, Figs. M. 15, 21, 22, 23 from diagrams on
52, 52, 34, 71. 84, 84, 90, 90, 115 ol the work respectively.
Andrade, Engines, Chap. VI, Fig. 5 ; Chap. TX, Figs. 1, 3, 4,
7, 8, IS, 16, 17 respectively from pages 239, Gl, 68, 75, 102, 86,
92, 107, 195, 211, 213 of the work by the kind permission of
Messrs. G. Bell & Sons, Ltd, (London) .
Partington and Shilling, Specific //eats of Gases. Chap. II, Figs. 13,
15, 17. respective:];, from pages 127, 76., B4 of the work.
OH, Theory of Heat. Chap. I, Fig. 3; Chap. II, Figs. 3, 12;
Chap. XI, Fig. 16, Reproduced by the kind permission ol
Messrs, Macmillan & Co.
Chapter
Pace
1. Thermometry
Temperature. Mercury thermometer. Special types of liquid
thermometers. Gas thermometers, Callendar compensated air
thermometer. Standard gas thermometers. Perfect gas scale,
.Jardization of secondary thermometers. Fixed temperature
Platinum thei era. Measurement of resistance.
mocouples. Low temperature thermometry. International
temperature scale. Illustration of the i i t Thermometry,
II. Calorimetry
Quantity of heat Methods in calorimetry. Method of mixtures.
Radiation correction. Specific heat of solids. Specific heat o£
liquids Method of cooling. Method of melting ice. BunserVs
ice calorimeter. Joly's steam calorimeter. The differential
steam calorimeter. Methods baser! on the rise of temperature.
oil of steadyflow electric calorimeter. Specific heat of
:ter, Results of early experi
■ arit. t. ion oi with temperature Two specific
. Experiments of GayLussac and Joule. Adiabatic
transformations. I ital methods. Method of mixtures,
method. Measurement of C. Explosion meti
Wiakitie i^xpansirn method). Experiments
of Clement and Desormes. Experiments of Partington. "Ruchardt's
experiment. Veloei und method. Kundt's tube. Expe
riments of Partington and Shilling. Specific heat of a
incur, Results, Special calorime 1
III. Kinetic Theory of Matter
The nature of heat Joule's experiments, I:. ■• "i in
mics, Metl ermrning /, !••!".. ai i nnciits.
i: ;>:: laboratory method of finding /, Electrical methods.
Heat as n I m decides. Growth of the kinetic tl
Evidence of molecular agitation. Brownian movement Pn
ted by a perfect gas. J : f gas laws. Introd
of temperature. Distribution of ■. I .'.veil's law.
cities. Law of equlpartition of kinetic energy.
■iilrir and atomic Mean free path phenomena.
:: of the mean tree path. Transport phenomena.*
Viscosity. Conduction, Value ol coi
[V. Equations of State for Gases
iation from the perfect gas equation. Andrews' experiments,
ai der WaaU* ei of state. Methods uf finding the
values of V and 'b'. Discussion of van der Waals' equation.
Experimental study of the equation of state. Experimental
determination of critical constants. Matter near the critical
point.
V. Change of State
Fusion. I a tent heat. Sublimation Change of properties on
ting. Determination of the melting point. Determination
of the latent heat of fusion. Experimental relationships.
104
VI
Chai
l*Afi»
lances at liquid
technique. Uses of
The air conditioning'
:. melting point. Fusion of
cooling, boiling and superheating. Saturated ami
unsati ■ lira Vaj <ur pressors of water. Statical
jethods. Dynamical method. Discussion of results, Vapour
pressure over curved surfaces. Latent heat of vaporization.
Condensation methods. Evaporation methods. Variation of
latent heat with temperature, Trouton's rule. Determination
of vapour density. Accurate determination of the density of
saturated vapour,
VI. Production of Low Temperatures .. ,125
Principles used in refrigeration. Adding a saft to ice. Boiling
a liquid under reduced pressure. Vapour compression inachine.
Refrigerants. Electric refrigerator. Ammonia absorption
machine. Adiabatic expansion of compressed gas ooling
due to Peltier effect. Cooling due to desorption. Liquefaction of
gases. Liquefaction by application of pressure and low tempe
rature. The principle of cascade,. Production of low tempera
tare by utilizing the JouleThomson effect Elementary heory
of the porous iperimeui. The porous plug experiment.
Principle of regenerative cooling. Lilidc's machine for liquefying
air. Hampson's air h'queficr. Claude's air liqueficr, Liquefac
tion of helium. Solidification of helium. Cooling produced by
adfabatic demagnetisation. Properties of
helium temperatures. Low temperature
liquid air. Principles of air eondi
hine.
VII, Thermal Expansion .. 157
ansion of solids. Lii rrlfcr measurements
of liner Camparatoi method,
ihod. Fizeau's interference method,
fringe ter. Surface and volume expansion,
Expansion of [ass, invar. Expansion of ■
bodies. Expansion of liquids. The difatometer method. The
weighty thermometer mctlmd. Matthiesscn's method, Absolute
expansion of Hqiuds. Hydrostatic balance method, Expansion
of mercury, water. Practical app
pensation of clocks and watches. Thermostats. Expansion of
Rases. t Experimental determination of the volume coefficient of
expansion. Experimental determination of the pressure co
efficient of expansion.
VIII, Conduction of Heat . . . . . . 178
Methods of heat propagation. Conductivity of different kinds
of maiter._ Definition of conductivity. Conductivity of mewls.
Conductivity from calorjmetric measurement. Rectilinear flow
"f heat. Mathematical investigation. EhgenHausz^s experi
ment. Experiments of Despretej Wiedemann and Franz. Forbes'
rcieUi I'om's metliod. Conductivity of the Earth's crust.
Conduction through composite walls. Relation between the ther
mal and electrical conductivities of metals. Heat conduction in
• dimensions. Conductivity of poorly conducting solids.
Spherical shell method. Cylindrical shell method. Lees' disc
method. Conductivity of liquids. Column method. Film method.
Conductivity of gases. Hotwire method, Film method. Results.
Freezing of a pond Convection of heat. Natural convection
VI 1
IPTER
IK. Heat Engines
Three different classes of jngines. Early history of the steam
engine, bfewcomen's atmospheric engine. James Watt. fJse
oi a separate condenser, The doubleacting engine. Utilisation
of the expansive power of steam. The governor and the throttle
valve. The crank and the flywheel. Modern steam engines.
Efficiency oi engines and indicator diagrams. The Carnot
engine. Reversible and irreversible processes. Reversibility of
the Carnot cycle Carnot's theorem. Rankhie cycle. Total heat
of _ steam. Internal _ combustion i engines. The Otto cycle,
1 scl cycle. SemiDiesel engines, The 'National' gas
engine. Diesel fourstroke engines. The steam turbine, The
theory of steam jets. The De Laval turbine. Ratcau and Zolly
turbines, Reaction turbines (Parsons}, Alternative types of
engines. Thermodynamics of refrigeratinn. Efficiency of a
vapour compression machine.
XI.
Page
204
X. Thermodynamics . .
.. 2c57
Scope oi thermodynamics. The thermal state of a body.
Mathematical notes Some physical applications. Different forms
of energy. Transmutation of energy. Conservation of energy.
Dissipation of energy. The first law of tiicmtodynamics.
Applications of the first law. _ Specific heat of a body.' Work
done hi certain processes. Discontinuous changes in energy —
latent heat Mess's law of constant heatsummation. Second
>f thermodynamics and entropy. Scope of the second law.
Preliminary statement of the second law. Absolute scale of
temperature. Definition of entropy. Entropy of a syg
Entropy remains constant in reversible processes. Entropy in
creases in irreversible processes. Entropy of a perfect ' gas.
General statement of second law of thermodynamics. Supposed
violation of the second law. Entropy and unavailable energy.
Physical concept of entropy. Entropy— temperature diagrams'
Entropy of steam. Applications of the two laws of thermo
dynamics, The thermodynamics! relationships (Maxwell). First
on. Application to a liquid film. Second relation. Other
itions. variation of intrinsic energy with volume, foute
Thontson effect. Correction of gas thermometer. Examples.
Clapeyron's deduction of the ClausiusClapevron relation, Speci
fic heat of saturated vapour. The triple point.
Radiation
Some simple instruments for measuring radiation. Properties
and nature of radiant energy. Identity of radiant energy and
hght— continuity of spectrum. Fundamental radiation processes.
Theory of exchanges (Prevost), Laws of cooling. Emissive
power of different substances. Reflecting power. Absorption,
itions existing between the different radiation quantities
Fundamental definitions, Kirchhoff's law. Applications of
Kifchhoff s law, Application to astrophysics. Temperature
radiation. Exchange of energy between radiation and matter in
a hollow enclosure. Deduction of Kirchhoff's law. The black
body. Radiometers! Sensitiveness of the thermopile. Crnokes'
radiometer. Bolometer. Kadiomicrometer. Pressure of radia
tion. Total radiation from a black body— the StcfanBollzmann
law. Experimental verification of Stefan's law. Laws of dis
tribution of energy in blackbody spectrum. Experimental study
ot the blackbody spectrum. Pyrometry. Gas pyrometers. Resis"
278
VIII
Chapter
Page
XII. Thermodynamics o> the ATSfosPHERE
Examples (I to XII)
Answers to Numerical Examples
Table of Physical Constants
Subject Index
316
327
331
349
351
352
CHAPTER I
THERMOMETRY
1. Temperature.— The sensation of heat or cold is a matter
t ail} 1 experience. By the mere sense of touch we can say whether
a substance is hotter or colder than ourselves. The hot body is said
to possess a higher temperature than the cold one.
But the sense of touch is merely qualitative, while scientific
ecision requires that every physical quantity should be measurable
numerical terms. Further, the measurements must be accurate
easily reproducible. This requires that the problem should be
handled objectively and tile sense of touch should be discarded in
favour of something which satisfies the above criteria. Let us sec how
can be done.
When two bodies are brought in contact, it is found that, in
general., there is a change in their properties such as volume, pressure
etc. due to exchange of heat. Finally an equilibrium state is attained
ter which there is no further change. The two bodies are then
said to be in thermal equilibrium with each other. In this state of
M, I equilibrium the two bodies are said to have the same tempe
which ensures their being in thermal equilibrium, Also it is
[ound that if a body A is in thermal equilibrium with two bodies B
and C, then li and C will be in thermal equilibrium with each other
therefore be at the same temperature. These are the two
fundamental laws of thermal equilibrium and it is on account of these
s that we arc able to measure the temperature of bodies B and C
bringing them successively in thermal equilibrium with the thermo
neter A. The temperature of a system is a property which determirii
whether or not a system is in thermal equilibrium with other systems.
Heat causes many changes in the physical properties of matter
some of which are well known, e.g., expansion, change in electrical
.laiire. production of electromotive force at the junction of two
iniilai' metals. All these effecLs have been utilised for the
isurement of temperature. The earliest and commonest therm o
titilise the property of expansion. Mercuryinglass is univets
em ployed as a thermometer for ordinary purposes, but though it
iiiplc. convenient to use and directreading, it is not sufficiently
for highclass scientific work.
Mercury Thermometer.** — Everybody is familiar with the
• • i ■. 1 1 1  . 1 1 ' centigrade thermometer. It consists of a glass bulb con
mercury to which a graduated capillary stem is attached.
The freezing point of water is marked 0°C and the boiling point 100°C
the interval divided into 100 equal parts. This scale was first
i In >1 by Celsiusf and is called the Celsius or centigrade scale
I r details of construction see Preston, Theory of FIcat, Chapter 2.
\ A i O us (17011744) was borti at Upsala where he studied matbe
tlci ii I astronomy. In 1750 he became Professor of Astronomy and ten years
hr built the observatory at Upsala and became its director. He invented
'"ale.
THERMOMETRY
and is now adopted for all scientific work. Other stales in ordinary
Use today are thos€ introduced by Fahrenheit and Reaumur. But,
Fahrenheit* was the first to choose mercury as die thermoinetric
substance cm account of its many advantages. It does not v,
can be easily obtained pure, remains liquid over a fairly wide range,
has a low specific heat and high conductivity ; it is opaque am
expansion is approximately uniform and regular. But we must not
orget its several drawbacks. The specific gravity and surface tension
of mercury are large, and the angle of contact with glass when
mercury is rising is different from that when it is falling. On account
of these defects alcohol is sometimes used in place of mercury, and
since it has a larger expansion it is more sensitive but. is likely
to distil over to colder pans oi the tube.
The range of an ordinary mercury thermometer is limited by
the fact that mercu: tes at 38.8°C and boils at 856 °C but
the upper limit can be raised to about 500 "C by filling the top of
the tube with nitrogen under pressure. The thermometric glass must
be of special quality ; it should be stable and should rapic urn
to its normal state after exposure Co high temperatures. The gla
generally employed are v .■• and Jena It'. 1 " for ' her
mometer; orosilicate glass 59 111 for hightemperature work.
Mercury thermomel generally o.d for rough work.
If tin at all fork various corrections must
be applied to get tJi: mpcrature. Tlie important ones am
ihe : —
This is very in po i ini
sine high tempers mal
ion due to change in the fundamental i om
to 100jS (say) I i lue.
ition Con his is due to want of uniformity
in the bore of the capillary n
(4) Correction for lag of the thermometer. This increases with
!ze uf the bulb,
(5) Correction due to changes in the size of the bulb caused by
variable internal and external pressure,
(6) Correction for the effects of capillarity.
Exposed stem correction. Part of the stem and hence the
contained mercury does not acquire the temperature of the bath.
For details concerning the application of these corrections
Appendix I.
3. Special Types of Liquid Thermometers. — The ordinary mer
cury thermometer is not suitable for certain purposes ; for this
reason special types of thermometers have been devised, are
"Daniel Gabriel Fahrenheit l b 161736) was born in Danzig of a rid fa
tmstenktn. He made improvements in the t
ised his '.. ric scale.
SPECIAL TYPES OF LIQUID THERMOMETERS
of the ordinary thermometer designed to serve the
\ lew,
cological purposes thermometers arc required to
! cate the maximum and minimum temperatures to which they
osed during a certain period. Six devised a combi
mum and minimum thermometer which is indicated in Fig. L
II if and part of die tube is filled with alcohol up to the level
in with mercui") up to C above which again there is alcohol
■ glass indexes I, I, have each an iron wire attached
(shown separately), and placed above B
Hid CI in each tube. When the tempera
es i lie alcohol in A expands and
landing mercury, on account of its
tension j pushes "upwards the index
ie C to its maximum limit. With a
I j II in temperature this index is undis
turbed due to the viscosity of alcohol being
II while the index above B is pulled
1 by the contracting alcohol, but is
I. n 'hind when the temperature rises,
iron wire attached to die glass in: I .:
prevents it from falling under its own
In. and enables its position to be
I From outside.
n ordinary
rem y thermometers oE
I the maximum type. The stem
the bulb has a constriction
■I which the mercury passes when
iis temperature rises. On cooling, the
iny is unable to force its way bark,
I lie range of temperatures is usually. 95 °F
i I I:) r and the bulb is very thin and
capillary bore very fine. The mercury
thread is rendered easily visible by con
. 1 1 1 1 « I i 1 1 g 1 ensf ron t thermomc t ers.
For accurate work, such as the determination of the boiling and
[ting points of organic substances, several shortrange thermo
• era employed between the range and 490°C. They are
U thermometers. Benzol and toluol thermometer
1 1 1 1 > 1 1 • ■ ■ i the many that are in use.
The Hechmann thermometer, indi I in fig. 2, is used to
ni> nail changes of temperature with a high degree of accuracy.
is here marked from to 5 representing approximai
centigrade degrees and even? degree is di into 100 equal parts.
.: reservoir at the top of the instrument, shown separately
H e the range to be varied. To set the thermometer lo
ii' desired range ihe bulb is heated to c n rcury into the reser
D A
• • ,■}']
:r.
Lib
:,..
*o
 '
IB
10 ' • ' V
1
10
'I
1  ,
•
OH
 16
i'B
■'••
' ' i""
!J
1 ■■'.■
ifi.
(\
E6
"j
1 r
go
"^^^
Fig. L — Six's maximum and
imum thermometer.
THERMOMETRY
[CHAIV
voir and the instrument gently tapped when the mercury column
breaks near the reservoir and some mercury i
transferred into it. Next the Beckmarm tin
meter is immersed along with an auxiliary
thermometer in a bath whose tempera tun
varied till the mercury stands at division "l
the former. The temperature corresponding to
the aero of the Beckmarm thermometer is thus
t I vi i v e 1. 1 on 1 1 1 e a uxiliary th ermom eter, and b)
varying the amount of mercury in the bulb this
is adjusted to be near the desired range. The
value of each scale division varies with the
quantity of mercury in the bulb and a correc
tion curve Lor different settings of the zero is
supplied with the instrument from which the
correction at any point of the scale ma) bi
obtained.
4. Gas Thermometers. — The fundamental
vantage of liquid them; sters is that two
thermometers containing differenl Liquids
as mercury and alcohol, and graduated as on
page 1 will agree probably only at.0°C and 100°C
ana at no other temperature. This is due to the
the expansion of the two liquid; is not
regular and similar, Thus the mercury thermo
page I would give an arbitrary* 6
of temperature. Moreover, the corrections to
iplied to it ■:}■■ 2) are uncertain and known
ly. Hence, tor accurate work
iry thermometers arc calibrated (see Sec. 8)
lual comparison with a resistance tin
meter throughout the entire range. Even then
the Riercur) thermometer is rarely used for
accurate work, and [or all standard work pas
mometcrs are employed.
Gases •• i/« I servants
i mometric substance. Their expansion is
large so that gas thermometers will be more
sensitive and the expansion of the containing
I will necessitate only a very small correc
tion, rheir expansion is also regular, i.e., the
nsion of a volume of gas at 0°C is the same
W
2< — Becfcmauu
thermometer.
•The v mercury at any temperature i lying
between and IQO''C as measured on the perfect gas scale is given by the r
Pi  Bd(1 + 1B182 x I0'.' + 078 X 10" t*)<
 the relation ts not linear and the readings of the mercury thermometer
iratod on p. I will not agree, even after applying the corrections mentioned
1 1 i A a perfect giv .. cter even in the range 0°— 100°G
1. 1 GAS rHURMOMETERS 5
cry one degree rise of temperature. They can be obtained pure
and remain gaseous over wide ranges of temperature. Further, the
.scales furnished by different gases are nearly identical since the volume
and pressure coefficients of all permanent gases are nearly equal.
Hence, gas thermometers arc used as primary standards with which
ill others are compared and calibrated.
The theoretical bases underlying the use of gas as thermometric
liiiii: :: are the laws of Boyle* and Charles which are very approxi
mately obeyed by the socalled permanent gases in nature but will be
rigorously obeyed b] a perfect gas. Let a gas be initially at pressure
/j,, volume V\ and temperature r t °C. TF we first change its pressure
from p! to p^, keeping the temperature constant, and next change the
temperature from t x to t 2 , keeping the pressure constant, we have
n 'in these laws,
. , ti' 1 + ox,
where v' is the intermediate volume and a the coefficient of expansion
at constant pressure, which is found experimentally to be approxi
mately equal to 1/273. Combining these two equations we have
Pi v i 1 " ' i
/V's
lffcf«
(1)
which is the gas equation.
II >'. [/a, ^2 = 0, U» at the temperature l/a = 273°C
i li ice point, the volume of the gas would be zero
provided the perfect gas equation is obeyed throughout the range,
This temperature is, by definition, called the ah &ro. It is true
ih: gas would liquefy and solidify long before this stage is reached
i in I the perfect gas equation would cease to be valid. Further, it is
inconceivable that matter should at any rime occupy no space (w=0).
Nevertheless l of such a zero of temperature is very useful. f
II we lure temperatures from this zero, the icepoint is given
by I/a, the steam point by (l/«)100, arid generally any tempera tun',
by (I /«) H 2 =7YK, Th c scale so obtained is called the Kelvin
• rale and will be denoted by °K, Hence equation (1)
i m omes
Pi"j
n
or
^2 J l '
where the suffix denotes the quantities at 0°C.
• ■ (2)
The quantity
■H Boyle (16271691) was b m in Trehnd but settled in England m
He distinguished himsdi in the study of Physio and Chemistry and was
" i the foundation members of the Royal Society. TTis main contribution is
I In' law of sases which still bears bis name.
1 I he absolute zero thus defined is shown from thermodynamic considerations
to be the lowest temperature possible. Hence the idea of this absolute
rry important [see further §71.
: iMOMETTRY
[chap.
. , T is known as the gas constant and varies as the mass of the
gas taken, but is approximately the sump for equivalent grammole
cules cl all gases. For one grammolecule this quantity is usually
denoted by R and is equal to 8.3 X ^ J " ergs/degree approximately.
If the mass of the gas is increased n times, the volume at the same
temperature and pressure will he increased n times and hence the gas
constant will also he increased n times. Hence, the gas equation cm
he. written generally in the form pv=nRT where n denotes the num
ber of grammolecules of the gas.
Equation (2) furnishes two ways of measuring temperature. The
pressure may be kept constant and' the volume observed at different
temperatures giving us the constant pressure thermometer ; or the
volume may be kept constant and the change in pressure noted, a
principle utilised in the constant volume thermometer. .
Exerctee. — Calculate the pressure of 20 grams of hydrogen inside
a vessel of 1 cubic metre capacity at. the temperature of 27 C.
[pv=?iRT where n=2Qf c Z. Ans. 0.25 atm.]
5. Callendar Compensated AirThermometer. — Accurate measure
ments with the constantpressure gas thermometers are difficult as
the gas in the connecting tube and
the manometer is at a tempera
different from that of the bulb. To
avoid this Callendar devised the
compensated air thermometer. In
tins instrument (Fig, 3) the pressure
of the aii in the thermometer bulb B
i the pressui e of the
air in D as indicated by the sulphuric
acid gauge G. When 11 is heated,
the pressure of the air in B increases
and equality of pressure is restored
by allowiifg mercury to flow out from
the mercury reservoir S. The volume of the tube connecting B and
S is eliminated by attaching to D an exactly similar tube placed close
to it. This will 'be evident from the following consideration: —
Let v, v it tfg be the volume of the bulb B, the tube conned
B and S P and the air in S respectively ; 6, lt a , their respective
temperatures, n the number of grammolecules of air contained in B,
S and the connecting, tube anil p its pressure, we have from the
gas laws,
.
*(t+i a*
Similarly if ?'', v/ be the volumes of the air in D and the tube attach
ed, 0', 0/ their respective temperatures, n f the number of gram
molecules and p' the pressure, we have
'{r+i)«
W
STANDARD fiAS 1 MEIERS
«', * =*/, h = &,', wc have from {3} and (4)
,i dition S and D are immersed in melting ice, a =0' =firf,
reezing , water, and
ine v tf, V 2 by B, D, S
B
= d °ns
(5)
Mi us we see that the influence of the connecting tubes is entirely
i a bed if (I) the pressure in B is kept equal to that in D (p — ff)\
i he total mass of the gas in B, S, and the connecting tube is
to that in D and the connecting tube ,{n = nf) ; (3) the volumes
Or tl ; ecting tubes
anal (vi — vf). The
i a nd i t i o il [di—8i)
is automatically satisfied
i e ilie two connect
tubes are placed
side by side and are at
i In same temperature.
6. Standard Gas
Thermometers. — T h e
i on slantpressure a i r
thermometer has been
nded by Callendar
ii . arious grounds : (1)
ih apparatus and the
. ulations are simple ;
i he internal pressure
on (he bulb does not in
, as the tempera
ture vises; (3) accuracy
of the results depends
i ii the accuracy of
hing. . tevertheless,
instrument dees not
give i : ■ i'. • rdant results
1 has been replaced
the constan tvolume
thermometer ns a stan
l. The normal ther
icter selected by the
Bute au 1 ntem a ti on al pig, 4 (a ) .— ■ Constantvolume Hydrogen Thermometer,
Poids et Mesures
and everywhere adopted today is the constant volume hydrogen tl\er
THERMOMETRY
[CHAP.
meter filled witli gas at a pressure of I metre of mercury at the
temperature of melting ice. It consists essentially of two parts ; the
bulb enclosing the invariable gaseous mass and the manometer for
measuring the pressure. Fig. 4(a) represents the thermometer dia
gr annua tieally.
The bulb C is a platinumiridium tube a litre in capacity, 1 metre
in length and 3G mm. in diameter. It is attached to the manometer
by a capillary tube of platinum 1 metre in length. The manometer
consists of two tubes A and B, and the stem of the barometer R dips
into A. The barometer tube is bent so that the
upper surface of mercury in it. is exactly above
\i and these levels can be read oft by a catheto
meter furnished with telescopes. The number
of observations to be taken is thus reduced to
two. B consists of two columns of mercury
separated by the stedpicce H and both these
columns communicate with A. By raising or
lowering the mercury reservoir M, the mercury
surface in the lower' part of B is arranged just
to touch a fine platinum point P [shown separ
ately in Fig. 4(6)], projecting from the steel
piece H, and thus the volume of the enclosed
gas is kept, constant.
The thermometer described above is suit
:: for measuring temperatures up to 600°G,
iperatures certain modifications
ary which will he discussed under
netry' in Chapter XI. The range of gas thermometers with
Ten irons ran be extended from 200° to 1600 "C.
We shall now deduce a Eormula for converting the observed
pressn nga into corresponding temperatures. T.I: /?, ; „ p vn denote
the pressure indicated by the manometer at the ice point and steam
point respectively, tlien p„v = riRTn, pimV — nR(T { 100) where T
represents the icepoint on the per Feet gas scale and the fundamental
interval is 100°C. Hence
to *v
rr ioo /to p >
where 8 is the coefficient of expansion at constant volume. Thus
we know ft for that gas from a measurement of p w and /_v To find
absolute temperature corresponding to any observed pressure p t ,
we have,
Tx ~Po
m
PERFECT GAS SCALE J
since T A = 1/8. Thus to determine an unknown temperature the
corresponding pressure *, is observed on the thermometer and he
temperature 1\ ^calculated either from the above relation or graphical y.
In an actual measurement corrections have to be applied for the
following : —
(a) The gas in the 'dead space' is not raised to the ^.nperatiire
of the bulb. The Mead space' consists of the space }»*" <^f
tube and in the manometer between the mercury level and the steel
piece H. Its initial and final temperatures are also different.
, Increase in the volume of the bulb C with rise of temperature.
(c) Change in volume of the bulb due to changes in internal
3S "d)' Changes intensity of mercury on account of temperature
changes,
For a discussion of these, the authors' book ',4 Treatise on Heat
may be consulted.
THERMODYNAMIC OR ABSOLUTE SCALE OF
TEMPERATURE
7 Perfect Gas Scale.— The formulae developed for the gas
thermometer assume that the gas, in question, accurately obeys the
E?as taws but experiments show that no real gas does so exact!), ine
coefficient of expansion « at constant pressure is not exactly the same
for veal eases as may be seen from any book of physical constants {see
Rave and Laby : 'PIMical Constants'). Further the two coefficients
* Fare not exactly equal, and also varies with the initial pressure
Thus, different gases would furnish different scales of temperature ij
the thermometer is calibrated as indicated above, and the selection
of anv particular gas will be arbitrary and will give an arbitrary scale
of temperature. To avoid this arbitrariness we must reduce our
ovations to that state of the gas.in which the perfect gas* equation
i, satisfied.
We shall now indicate the mediodsf of reducing the observations
on real gases to the perfect gas state. This can be done when we
have knowledge of the deviation of gases from Boyle s law. 1 he
calculations are rather complicated and will not be given here. It is
enough to point out that the equation for any real gas can be written
in die form
PV. z=.RT + Bp +\Cp* + D / ,a + '
where! B, C, Z>...aie constants which go on decreasing rapidly. Thus
ii is evident (hat at infinitely low pressure (p*0) all gases will obey
Boyle's law accurately and this conclusion is borne out by expert
* A perfect gas is defined as one which will obey Boyle's law and Joule's law
(Chap. II. sec. 22) rigorously.
t For fuller information sec A Treatise ott Heat by the Authors.
J Further see Chap. IV, Section 1.
10
II!' I!. MOM IMR1
[t KAP,
rnent;. ation. Now it is mentally found that the tempe
rature scales, obtained by using the different gases and extrapolating
observations to zero pressure, art: actually identical lor all the
This is the perfect gas scale. The coefficient of expansion «
for a perl: an be calculated in this manner from the data given
by Heuse and Otto. The mean of several results gives a = 0.00
for a perfect gas. Thus T the melting point of ice on the perfect
gas scale— l/« = 273.10°K, Considering all the available data Birge
adopts the value 273.16^0.01. We shall use the value £7::
or 273. pending upon the degree of accuracy required. Aunin
i ;j accurate experimental data on oxygen we have
Lim (
22.411 litre X atmospl
Hence
fl = Lim (pV)„
22.414 x 10 a X 76 X 13.595 X 081 .,
97 * Tfi ergi/degree
273.16
= 8.314 X I(T :
1.186 X ' 'I' 11 ' 1 ' III), hence R=l
, per degree. This is the value of the gas constant. For real
alue of the quantity p G f',,/7',, differs only slightly From this,
v point out i alt arrived at in Chaj
tvn how Lord Kelvin, of heat
scale independent of the property
scale of temperature
the thermodynamic scale. Further it is shown there
i uite identical with the perfect gas scale. We thus
see that th< . scale which was hitherto shown to depend
on the properti; is now becomes independent of the properties,
any particular substance. Hence it is called absolute scale
standard scale adopted in scientific work.
r method of obtaining the correction to be applied to the
real gas scale consists in performing the JouleThomson experiment
(Chap. VI). But unfortunately the existii g data on JouleThomson
effect are no tent to enable us to apply this method and the
method given above is almost universally employed.
S. Standardization of Secondary Thermometer*. — Gas the^ntome
• are very cumbersome to use and require several corrcctio
Hence in labor a to are replaced b; secondary standards, such
as the resistance thermometer, the thermocouple, etc., which have
been carefully standardized by comparison with a standard gas thermo
er in standardizing laboratories like the National Physical Tab.
or the Bureau, of Standards in Wa
RE BATHS
I I
pur] imparison baths may be constructed, each suitable foi
particular range. Between 0° and* 100°C a water bath, between 80°
and 250°C an oil bath, between 250° and 600 q C a mixture of potas
sium nitrate and sodium nitrate., and above that an electrical heater is
generally employed,
The secondary thermometers may also be standardized by means
of a series of easily reproducible fixed points whose temperatures have
been accurately determined. A table ot standard temperatures is given
below (Table I.) . The values are generally those adopted by the
r enth General Conference of Weights and Measures represi
thirtyone natrons which was held in October 1027, but some amend
ment, made by the Ninth General Conference in 1948 have also been
incorporated.*
9, Fixed Temperature Baths. — It is frequently convenient to
calibrate the secondary thermometers by means of the fixedpoint
scale given in Table 1. The icepoint may be most conveniently
obtained by dipping the thermometer in pure melting ice contained in
a dewar flask. This is a doublewalled glass or metal vessel whose
sides are silvered.]* For the steampoint the hypsometer indicated
in Fig. , r >, p. 12 is employed. The diagram explains itself. C is the
v I, — Standard Temperatures.
Temper?' i
Substance
Temperature
Substance
Centigrade
tvti nade
252.780°
E. P. of Hydrogen
419*5
F. p. of :
195
B. P.
444 vi
B. P. of Sulphur
 182
:. P. of Oxygen
630 5
F. P. o : Anil :i
 7S5
Sublimation of CO»
9m 8
M. P g Ivor
 3887
F. P. of Mercury
1063
M. P.
000
:■,:. p. of ke
F. P. of Coppi
+ 3238
Transition temperature
1453
U. P. ol Nickel
of Na.SOaOH.0
1553
F. P. of Palladium
iooorjo
B. P. of Water
1769
F. P. of Platinum
2180
B, I ■ > hLbalene
2443
F, P. of Iridium
2319
R P. of Tin
2620±10
M. P. of Molyb
3059
B. P. of Bcnzophenone
denum
3209
F, P. of Cadmium
3380±50
M. P of
3273
F. P. of Lead
3500+50
M. P. of Carbon
*See "The International Temperature Scale of 1948", National
Laboratory, Teddington (1949),
tFor a complete description see Chap. VI.
32
THERMO M KTRY
[chap,
condenser employed lo present the water from being lost by evapora
tion, M is the manometer and T the thermometer. The path of steam
is indicated by arrows. The boiling point of water at the pressure p
(in mm. of mercury) h found to be given by the relation
* = 100.0001S.67 X I0~ 2 (p  760)  2.3 >< 10* (p  760)2.
For other fixed points a number o£ vapour baths in which sulphur,
naphthalene, aniline etc. are used, serves the purpose,
_ For determining the boiling point of sulphur Callcndar and
Griffiths found that the standard Meyertube apparatus was verv
suitable. It consists of a hard glass cylinder A of diameter 5 cm.
and length about 25 cm. to which a spherical bulb B is attached at
the bottom (Fig, 6) , The whole is surrounded by an asbestos chamber
C. The thermometer T is fitted with an asbestos or aluminium cone
Fig. 5, — Hypsometer.
6. — Sulphurboiling apparatus
D. This cone serves in Lwo ways : (1) it prevents the condensed
sulphur from running down over the bulb and cooling it below the
temperature of sulphur vapour ; (2) it prevents the bulb from directly
radiating to the cooler parts of the tube. Sulphur is placed in the
bulb and heated aver a flame. A side tube may be provided in the
upper part of the chamber and serves to condense sulphur vapour. The
boiling point of sulphur is given by the formula
t = 444.60+9.09 X 1«~ 8 (P 760) 4.8 X 10 •"(/> 760)*.
PLATINUM THERMOMK'iKKS
13
CP c
Baths for naphthalene and aniline may be constructed by slightly
modifying the above apparatus.
RESISTANCE THERMOMETRY
10. Platinum Thermometers.— The necessity of
, ndary standards has been clearly indicated above.
:> tvpes of such instruments based on two electrical
properties of matter will be described in this chapter.
These properties are— (1) variation of electrical resis
tance of metals with temperature; (2) variation of
thermal electromotive force with temperature. First
let us consider the former.
Sir William Siemens was the first to construct
ermometer in 1871 based on this principle but^he con
structional details were unsatisfactory. Later improve
ments by Callendar* and Griffiths have given the instru
ment its modem form. Fig. 7 represents an hermetic
allysealed thermometer designed by Dr. E. H. Griffiths
for laboratory work of high precision. Pure platinum
ire free from silicon, carbon., tin and other impurities is
: ted. It is doubled on itself to avoid induction effects
and then wound on a thin plate of insulating mica m.
The ends of this wire are attached to platinum leads
which pass through holes in mica sheets closely fitting
the upper part of the tube, and the other ends of these
ads ire joined to terminals P, P at the top of the instru
ct. The mica sheets give the best insulation and
vent convection current of air up and down the tube,
L e coil is sealed for, otherwise, moisture would deposit
'in the mica and break down the insulation. To cpm
iisate for the resistance of the leads, an exactly similar
pair of leads, with their low ends joined togeth
placed close to the platinum thermometer leads, and is
, umected to terminals marked C, C, on the instruir,
hese are called compensating leads and are joined in
the third arm of the Whcatstone bridge as shown in
r, 8 (p. lft). Then since the ratio arms are kept equal
;uid the compensating and platinum leads have equal
resistance at all temperatures, it is the resistance of the
inum coil alone which is determined. For work up
to 700 °C copper leads may be used and the whole may
l e : nclosed in a Lube of hard glass. But for high tem Fjg 7^pi ati .
peratures platinum leads must he used and the whole num. therm
be enclosed in a tube of glazed porcelain. meter.
The precision and reliability of modern resistance thermometers
ate entirely due to the work of Callendar and Griffiths. They deter
* H. L. Callcndar (18631930) was educated at Cambridge and worked it; the
I •, tidtsh Laboratory from ISSS to 1890 on resistance thermometry. His greatest
are the development of the platinum resistance thermometer and the
Investigation of the properties of steam.
M
MOMETKV
MEASUREMENT Ol? RESISTANCE
.
latmum from 0° to 500*C and found
'" ,L Itwas ™*y a ^n fay a parabolic formula of the
= J2 «( 1 B*)* ■ . . . (7)
C and 0»C and .
, ,,..9.J lO» f = _$8 • I0 : ■ it ,, i
' ' li « i .. Srytosoh q B u U a
£c equation to find  from the value ol [ e „ ave '
*™ tothl ! lario ,P™ ducednomeE
1,111 PWrt ";"• Thus we define the platn
ans of the simple linear relation* Kinpen , t p by
m
''fc=S ,o °
US' SiS
ill 00/ 'loo' • •
*ri?
100
100a ( (100) 3 /?
_ £(100)* r/ mi i ]
a+ 100^ H 100/
Thus S in equation (9) is equal to
x loo > /
 !0Q£*
tie specimens emploved is nhnm i r, T t,
•« I ' Ufa different from n . The exact relation may he fcduc,
4
finding the platinum temperature ^ lor the boiling point of sulphur
temperature is known, and then substituting in (
Kir i tor a specimen, use my unkno
ature. The the thermometer at the unkno
temperature is found out and ; ( tf ) h ** determined Froi
this, using the value of 8, the true temperature i can he determined
will jip of (9). Il will be ol that the correction i
giving the value of tt t ins the unknown temperature t.
voik the value ol t f may be substituted for t on the righthand
(9) . Tor accurate work, however, the procedure is as follows : —
Tin hand side of (9) is calculated for different assumed values
ol I able is constructed giving the value of this correction t< I
different values of tp. With the help of this table the true tem
perature i corresponding to the experimentally determined platinum
tempera i in r t p is found.
It was shown later by Heyoock and Neville, and Waidner and
Bin i if die platinum thermometer is standardized at Q", 100°
boiling point o( sulphur the parabolic formula (9) gives true
L! Ear as 630 °C
We shall illustrate the method by, 'a numerical example. Let
the resistance ol a given platinum tliermometer at 0°, 100° and
i 4.6) be 2*56, 8. 56 and 6.78 ohms resj
equired to calculate the true temperature when the
resis ' E the. thei mometer is 5.56 ohi
U 5 ,' 5 . e X 100 = 3GG°C.
t,
"~ 35G256
for the boiling point of sulphur
_ 678.
" 356
4446422 =S
7, *
100 ^ 422°C.
1 
1 1 id / 100
hence & = 1.4.
From (9) we get / = 300, (*,== S.I; i p = 291.2°,
t — 320, tt p = 10.4 hence f. = 309.6°.
►rrection tt p for t f = 291.2° is 8.8 and for 309.6 it is 10.4.
: 300 it is B.B + ~=^X (300291.2) = 9,6.
rherefore the true temperature t is equal to 309.6
11. Measurement of Resistance. — The determination or tempera
. this thermometei Involves the accurate measurement ol the
ace of the platinum wire. Various special types of resistance
re used lor this purpose. In order to compensate for the
ice ol the leads, a bridge with equal ratio arms i used. Fui
the brii g< should be capable of measuring cbaxi e to a
ree of accuracy & fundamental interval is
> and measurement of temperature to hundedths of a tl
requi stance measurement to one tenthousandth of an ohm.
16
THitRMOM ITRY
CHAP.
Fig'. 8j— Calliflniaratid Griffiths Bridge.
adjusted for no deflection of the galvanometer G.
Then
The Callendar and Griffiths bridge* is quite suitable for this pur
pose. Fig, 8 indicates r.he connections. Q and Si the ratio anas,
arc kept equal by the makers of the instrument. R consists of a set
of resistances of 1, 2, 4, 8,' 16, 32,
64 units. The usual plug contacts
are here replaced by mercury cup
contacts. L x and L. axe two paral
lel wires of the same material
which can be connected to each
other by the contactmaker K,
This arrangement is adopted in
order to eliminate thermoelec.tr o
motive forces. P represents the
thermometer and C the compen
sating leads. The resistance r
acts as a shunt and makes the
resistance of the wire exactly in
the desired ratio. The bridge is
Suppose the
balance point is obtained with the key K at a distance x from the
centre of the wire and die entire length of the wire is 2a.
or P =r.R — 2x P .
where p is the resistance per unit length of the wire. It is found
it to select the wire L 3 and che shunt r in such a mannei that
1 cm. o£ the bridge wire has a resistance of 1/200 ohm and the total
of the wire is 20 cm., while the smallest resistance in R
d 1 has a r: >. d stance of tfie whole
i.e.,, 20/200 — 0.1 ohm. Thus if the fundamental interval of the
it'Kc thermometer is 1 ohm the temperature can be determined
C provided the balai at is determined correct to
Oil mm. For accurate work., however, various precautions are neces
which are given belowj ; —
The current, flowing through the bridge heats the bridge
coils and changes their resistance. The change in tem
perature may be observed on a thermometer and the
corresponding change in resistance calculated. The correc
tion can then be easily applied. Or the bridge ni;i
placed in a thermostat.
The thermometer coil has to be very thin (0.15 mm. dia
:,Y) since it must have a large resistance and hence
the heating effect is considerable. From Calendar's
observations the heating effect for a current of .01 ampere
is 0.016° at 0°C and 0.017° at 100 C C. According to him
the best, procedure is to pass the same current through
* For further details see Flint and W'orv.oj 'I'mciiaii I'kys'u.;'.
i For full details see Methods of Measuring Temperature by E, Griffiths,
(Chap. 3.)
(2)
B i 1: A Sl.I REMENT OF RESIST A 
17
the thermometer at all temperatures when the heating
effect remains approximately constant.
(#) The bridge centre must be determined and the bridge wire
calibrated.
(4) Due to temperature gradient along the conducting leads
and the junctions thermoelectromotive forces are developed
in the circuit whose magnitude may be found by closing
the galvanometer circuit when the battery circuit is kept
open. To eliminate these the galvanometer circuit should
be permanently closed and balance obtained for reversals
of the battery current. If induction effects are perceptible
when the battery * circuit is made or broken a thermo
electric key should be employed. This key first breaks the
galvanometer circuit, then 'makes the battery and the
galvanometer circuits in succession.
(5) The external leads connecting the terminals PP, CO to the
bridge should be exactly similar and similarly placed.
12. As already mentioned the platinum thermometer is standar
dized by measuring the resistance at the melting point of ice, boiling
points of water and sulphur. The last gives S and hence t can be
determined from any subsequent determination of t p . Eqn. (9) how
ever does not hold much above 630°C and therefore every such thermo
meter is provided with a calibration curve drawn by an actual com
parison with a standard gas thermometer in standardizing laboratories.
The temperature can be directly read from this tempera Weresistim ,e
curve,
The great advantage of platinum thermometers lies in their wide
range (~20Q°C to 121KPC). If carefully prepared, their readings are
reliable to 0.01° up to 500°C and to 0.1° up to 1200°G but generally
it is not desirable to use them above 1000°C owing to the dan :
contamination toy the insulating materials. They are free from
changes of zero for the wire when pure and wellannealed has always
the same resistance at the same temperature. They are very con
venient for ordinary use n.nd, when onee .standardized by comparison
■ ith a gas thermometer, they serve as reliable standards. They are
employed to measure' small differences of tempera tit re very
ately, sometimes even to one tenthousandth of a degree. There
re, however, some drawbacks also. The resistance thermometer
has a J: tge thermal capacity and the covering sheath has a low thermal
Conductivity and therefore the thermometer does not quickly attain
ilt: temperature of the bath, in which it is immersed. Further some
i ! Lost in balancing the bridge. For these reasons the resistance
lometer is useless for measuring rapidlychanging temperatures.
urther impurities in the platinum do not obey the same resistanre
I mperature law as the pure metal.
I 2 gives the variation in resistance of a platinum therm o
metei over a wide range. It is taken from Henning's Temj
. The value of the quantity R = R t /R is given for various
o
i;,
THERMOMETRY
[chap.
temperatures where H„ R are the resistances at tempera tine:.
m! ii ( ! respectively.
Table 2,— Values of R—R t /R^
Ten: .
R
Temp,
R
Ten
J?
D C
021607
°C
m
+40
1.15796
280
2.06661
180
0.25927
60
1.23624
300
2.13931
160
0.34505
80
1,31406
320
2.21154
140
0.42986
100
1.39141
340
2J28330
120
0.51347
120
146830
360
2.35460
100
O.S9i
140
1.54471
2.42543
 80
0.67814
160
1.621
400
2.49S80
 60
0.7594$
180
1.69616
420
.570
 40
200
1.77118
440
2,63513
J 20
0,92033
220
1.85474
•li:i
2.70410
1.00000
240
1.91983
480
2. 77261
20
1.07921
260
1.99345
THERMOELECTRIC THERMOMETRY
13. ThermoCouples. — Let us now return to the second electri
asurement. Starting from
■ ., in 1821 nm ■• were made to
ometer based on this principle, Eor instani
Pouille( and Regnault. At present thermoelectric
attained a degree of precision inferior only to
rice thermometry below 1000°C, but foi atrires exceed
::ic it [g the onl) sensit: I convenient electrical method
at our disposal.
A thermoelectric thermometer installation consists of the follow
ing parts :—
;]; The two elements constituting the thermocouple.
' The electrical insulation of these wires and the protecting;
tubes.
(3) Millivoltmeter or potentiometer for measui in the thermo
electromoi h e force,
(4) Arrangement for controlling the coldjunction temperature.
The choice of the elements constituting the couple is determined
tperature to which the couple is to be heated and .the e.m.f.
developed. For low temperatures up to 800°C couples of base mewls
such as ironeon sf a ntnn and copperconstantan are satisfactory, as they
lop a large e.m.f. of about. 40 to fif) microvolts per degree. For
H atures these base metals cannot be used as thev get
ized and melt. Nickeliron couple may be used up to 600° while
nickelnichrome and chromelalume) thermocouples can be used up
']
THERMOGOUP LES
19
to 1000° but above that platinum and an alloy of platinum with
iridium or rhodium must be used. Le Chatetier in 1886 introduced
the couple consisting of pure platinum and an alloy of
90 per cent Pi and 10 per cent Rh which is no
employed for scientific work. The e.m.f. developed by
these noble metals is, however, much less.
The two elements are taken in die form of a wire
aud one end of both is welded together electrically or
in an ox y hydrogen flame. This end « (Fig, 9) forms
the hot junction. The portions of the wires near the
hot junction are insulated with capillaries of fire
clay (or hard glass for lower temperatures) and are
threaded through mica discs enclosed in outer protect
ing tube of porcelain, quartz or hard glass, depending
upon the temperature for which it is meant. The
protecting tube prevents the junction from contamina
tion but necessarily introduces a lag. For rough use
this may be further enclosed in a steel sheath (shown
black in the figure) . Where there is no risk of conta
mination, the mica discs and the protecting tubes can
be dispensed with. The wires are connected to terminals
d ana! C s on the instrument. To these terminals are
connected flexible compensating leads leading to the
cold junction ' (Fig. 10a) . These leads are usually of
the same material as the elements of the couple itself.
Thus the cold junction is transferred to a convenient
1 1 hi mt place where a constant temperature, say Q C,
can be maintained. Usually the compensating leads
Hi; marked so that there is no difficulty in connecting
jo the proper terminals.
There are two ways of making the connections
which are indicated in Figs. 10 (b) and (c) . The dia
grams explain themselves. The cold junction is
immersed in ice at 0°C. As a recording instrument
r a millivoltmeter or a potentiometer is employed.
a
Fig. 9 —
Thermo
couple.
i i couple
Cold
JtUlCtiun
Hat
Illl'i til I
Cotnpeasating
extension
Hot
junction junction junction
W (6)
Fig, 10.— Measuring with a tincnnocouple.
:v:
14. To Unci the temperature of the hot junction we must
tire the e.m.f, &< bei « een thi enda ' : me o ►pper leads.
20
THERMOMETRY
[CHAP.
I li : s ran be done by means of a high resistance miilivoltmeter which
mav be graduated to mperatures directly and die temperature
thus obtained can be relied
upon to about ±$°C. For
attaining higher accuracy
Fig. H.— Illustration of the principle
of potentiometer.
a potentiometer must be
used. This arrangement
essentially consists of a
number of resistance coils
A (Fig. 11) placed in series
with a long wire resistance
r stretched along a scale.
A current from the battery
E flows through these re
sistances and its sti:
is so adjusted by varying
R that the potential differ
ence across a fixed ■
ance X balances against the c.mX of a standard cadmium cell C
{L0183 volts) . The e.m.f. developed by the thermocouple Th is
balanced as indicated. The potentiometer can be made direct reading
bv keeping K=z 10 LP ohms. Thus there is a fall of 1 volt pe
ohms and by constructing; the smallest resistance coil of fl.l ohm
mce and the wire r also of the same resistance, the total e.m.f,
: the wire will be 1 m.v. If the wire is divided into 100 d.iv
and in addition lias a sliding vernier having TO divisions the readings
)e taken correct to 1 microvolt.
Various types of potentiometers based on this principle have been
devised specially lor this purpose* With these insiniments the
e,m.f. can be measured accurately to 1 microvolt which corresponds
!,, r a pt— PtUh couple. For a copperconstantan couple
this corresponds to about 1/40 degree. With a sensitive arrangement
it. is possible to measure to 0,1 microvolt when the sensitiveness is
increased about ten times. For accurate work the cold junction must
be maintained at 0°C otherwise corrections! will be required in that
respect.
In order to deduce the temperature from an experimental deter
mination or the e.m.f. a calibration curve is generally^ supplied with
the instrument. This gives the temperature corresponding to different
electromotive forces developed and has been drawn by the makers
by an actual comparison with a standard thermocouple Throughout
the range. If it is required to calibrate a thermocouple in the
absence of a standard one, the fixed points (Sec, 8) must be utilised.
The e.m.f. at those points is measured and an empirical interpolation
formula employed in order to give the e.m.f, corresponding to the
* A description of these will be found in Methods of Measuring Temperature,
by E, Griffiths.
f See Ezcr Griffiths, Methods of Measuring Temperature, (1947), p. 74.
THERMOCOUl'LhS
21
rmediate temperatures. For a Pt— PtRh couple three different
equ: must be used for the different ranges. Thus
; am 0° to 400*0, E = At + &(l O ,
300° to 1200 C C, E =  A* + & + &?,
1100° to 1750<>C, E = A" + B"t f C"l*
where A, B, C are constants whose values are empirically determined.
Thermocouples are frequently employed for laboratory work since
v are cheap and can be easily constructed. They can be used for
i measurement of rapidlychanging temperatures since the thermal
city of the junction is small ami hence the thermometer has
. ctieally no lag. Another advantage in the use of thermocouples is
they' measure the temperature at a point—the point at which
the two metals make electrical contact. Its chief disadvantage lies
the fact that there is no theoretical formula which can be extra
polated over a wide range and consequently every thermocouple
requires separate calibration.
The useful range of thermoelectric thermometers is about  200 &
to 1600°C. Readings are reliable only, when the composition of the
i ile does not change even slightly. In actual practice frequent
calibration is necessary.
The following are the chief sources of error in thermoelectric
i mometry : —
(1) Parasitic electromotive forces developed in the circuit. They
are due to (a) Peltier effect or e.m.f, developed due to heating of
. t.i.ort of dissimilar metals at points of the circuit other than the
i and the cold junctions. This occurs often in the measuring
>aratus ; (b) Becqucrel effect or e.m.f, generated due to inhomo
geneities in a single wire; this occurs mainly in the thermocouple
wires. The e.m.L measured is a sum of these quantities and the
Peltier e.m.f* at the two junctions and the Thomson e.m.f, along
homogeneous wires of the thermocouple with ends at the two tempe
ratures. The undesirable effects mentioned in (a) and (b) must be
dilated by the use of materials and methods free from these effects
nee they are not taken into account in any thermoelectric formula?.
(2) Leakage from the light mains or furnace circuit. If leakage
s through the potentiometer their presence can be detected
1 1', shortcircuiting the thermocouple when the galvanometer continues
to be deflected.
I) Coldjunction correction if it is not kept at (PC.
For the methods of minimising or eliminating these errors the
is referred to Measurement of High T wres by Le
Chatelier and Burgess.
Table 3 compiled from various sources gives the thermoelectric
J. for various couples in common use. The cold junction is
• i, lined at 0°G* and the hot junction at t°Q. The e.m.f. of the
'•nocouple AB being positive means that the current flows from
\ to P. at the cold junction.
22 THERMOMETRY [CHAP.
Table 3. — E.m.f. in Millivolts for some thermocouples.
Temp.
90Pt10Rh
.'Ni
Ag/Pl
Fe/cons
Cu/cons
fC
against Ft

tarrtan
tantan
200
S.27
.
100
■
3.349
 80
— 1.68
0.30
r;.co
0.00
;:i::i
0.00
0.000
+ 1
+0.643
+5.18
+0.72
I 5.40
+4.276
200
1.436
.96
+1.73
10.99
9.285
m
2.315
752
2.96
16.56
14:'
400
3. 250
9.83
4.47
500
4,219
12.04
6.26
27,58
l.llii
5.222
1450
8.25
700
S.260
17.30
39.30
800
7.330
20.73
13,17
45.72
000
8.43
24.19
15.99
9.569
.
US
14.312
1600
16,674
17 Ml
15. in methods of measuring temperature utilise the radia
; to he measured.
will be discussed in detail later (sec Chap. XI). "J
I for measuring temperatures from about lOOO^C to any
upper limit.
16. Certain other methods of measuring temperature utfljse any
one of the following properties of matter : —
(1) Expansion of a bar of metal.
(2) Changes in vapour density with rise of temperature.
(3) Variation of refractive index oF a gas with temperature
accordance with Gladstone and Dale's law.
": Calorimetric methods based on the measurement of quantity
oi h
: Change of vapour pressure with temperature.
17. Low Temperature Thermometry.* — The standard thermo
meter in this range is the constant volume hydrogen or helium thermo
meter. The difficulty in this case is that gases liquefy and even
lify at these low temperatures. Prof. Dewar, however, showed
that the boiling point of hydrogen as indicated by the liydrog
thermometer was 253.0 o C and 253.4°C, while a helium thermc
meter registered 253.7°C and 252.1°C, Similarly he compared
* For pyromelry see Chap. XT.
LOW TEMVl'UATURE THERMOMETRY
23
thermometers of other gases. These experiments evidently led to
the conclusion that a gas could be relied upon almost to its boiling
point. Thus helium furnishes the scale down to its boiling point
(4.2° K.) . The corrections necessary to convert this scale to the
thermodynamic scale may be obtained and have been given by Onnes
and Gath. For temperatures below 4.2°K we must use the helium
gas thermometer with die pressure well below the vapour pressure
of liquid helium at the temperature to be measured, so that the gas
will not liquefy at that temperature. With this device the helium gas
thermometer gives us the thermodynamic scale down to 1°K. Now
we shall consider the secondary standards.
Mercury freezes at 38.87 and alcohol at 1]1.8'"C and hence
these thermometers cannot be used below the respective tempera
tures. A special liquid thermometer containing fractionally distilled
petroleum ether can be used down to  190*0,
But for all accurate work, however, resistance thermometers are
employed. It is absolutely essential that the substance of which the
thermometer is made is perfectly pure. Pure metals show a regular
rise in resistance with decrease of temperature. Dewar and
Fleming found that the presence of the slightest trace of impurity in a
metal is sufficient to produce a considerable increase in resistance at
these low temperatures. It is, therefore, difficult to trust the purity
of any specimen for very low temperatures without actual comparison.
Henning found from a detailed investigation that the parabolic formula
did not hold below — 40 D C. Van Dusen proposed the formula
where e is a constant and 8 has already been defined on p. 14. The
constants, R lf! K im and 8 are determined by calibration at 0*0,
mil c; and the boiling point of sulphur as explained previously, and
the constant e is then determined by calibration at the boiling point
of oxygen ( I82.97°Cj . Van Dusen s formula has been found to hold
atisfactorily from 0°C to 190°C, die error nowhere being greater
ban ri=0,05 a .
For temperatures lower than  190°C, the platinum thermometer
was used by Henning and Otto, and can be used with advantage up to
20°K. There is, however, no satisfactory, formula for calculating the
temperature from the observed resistance and a calibration curve has
to be used. Sometimes lead and gold thermometers are also employed.
Onnes has used lead down to 259°C and Nernsl lias, given a method
dculating these temperatures. Below —250*0 resistance ther
mometers of cons tan tan and phosphorbronze have been employed, the
latter being much more sensitive.
For low temperatures coppercons tan tan and ironcons tantan
couples are very sensitive as they develop a large e.m.f. They can
be used down to  255 B C. They must be calibrated by direct' com
Eson with a gas thermometer.
^4
THERMOMETRY
In order to measure temperatures below the temperature ol
helium (268°C) the vapourpressure thermometer o£ helium can be
employed, Its use is based on the wellknown fact that the vapour
pressure of a liquid varies uniquely with the temperature. Thus the
method consists in measuring the vapour pressure of a liquid at the
required temperature by means of an apparatus similar to that shown
in Tig. 4, Chap. V, and obtaining the corresponding temperature by
means of a calibration curve or a theoretical formula. The helium gas
thermometer and the vapourpressure thermometer have been used
down to about 0.75 C K. For measuring still lower temperatures the
paramagnetic susceptibility of salts is utilised.
18* International Temperature Seal*.— We have seen that the
thermodynamic centigrade scale is the standard scale of temperature
d is given by the helium gas thermometer, but gas thermomel
involves many experimental difficulties. On account of these difficulties
in the practical realisation of the thermodynamic scale the Inter
M iiovial Commitee in 1927 found it expedient to adopt a practical
scale known as the International Temperature Scale. This scale agrees
with the thermodynamic scale as closely as our present knowledge
permits and is at the same time designed to be easily and. accurately
reproducible. It is based upon a number of reproducible fixed points
to which numerical values have been assigned and the intermediate
temperatures have been defined by agreement as the values given
by trie following thermometers according to the scheme given below ; —
(Pi 9 1 and and platinum resistance
neter calibrated at D , I00°C and the boiling point
sulphur.
: From 190 platinum resistance thermometer
which gives temperature by means of the formula
Et = R (I f <xt + fit 2 + y («  100) **},
the four constants being determined by calibration at ice,
steam, sulphur and oxygen points. It, will be seen that i.iv
formula is equivalent to (10) where e s= (I00) 2 87//J.
(3) From 661>°C to I0b"S°C.— The platinum PtRh thermocouple
where temperature is defined by
E = a f bt 4 cF r
and the three constants are determined by calibration at the
dug point of antimony and at the silver and gold points.
(4) Above 1063°C— An optical pyrometer (see Chap. XI) cali
brated at the gold point (106S D C) .
it should be emphasized that the International Scale does not replace
the thermodynamic scale ; it merely serves to represent it in a practical
ter with sufficient, accuracy for most purposes.
>•]
0«K
lOO'K
273K
IflO'C
200
300
m
SCO
6Q0
TOD —
SOD —
: :o
:;. ::
CHART OF FIXED POINTS
Ch art of Fixed Po ints
iSiOLi'. B. P. of Helium
283.7E 1 B, V. of HydrDgeu
LS5'6J B. P. of Nitrogen
^UgS«a B P. of Oxygen
 784E3 Sublimation of CO*
 ss632 F. P. of Ifenury
D'OU Water freaes
• 9S'3S4 TroBsSttonofNa^lOHjO.
■ lOftOtt B, P. of Water
2?)
419 '43
4,14 '60
SITS* B. P. of. Naphthalene
231*86 P.P. of Tin
30590 B. P. of Bensoptienmic
3209 F . P. of Cadmium
1 r 
B. P. of Sulphur
' .•■ .:, ' . .
523*C. Draper point
iBed Just
Visible
SKI'S F . P. of Antimony
600' I F. E of Alurnjniym
m9, F, P. of Silver
Dull red
Cherry
red
Bright
□harry
1
&
■;
'
26
i ."I
i
I'I.M
liO'J
!':'•'
2400
SWO
THFRMOMKTKY
Chart of Fixed Points (Contd}*
[chap.
M. P. of Gold
p.P.ofCopJn:r
■1*53 F.P. ofNkNri
■15S2 F.P of Palladium
176& F. P. of Platinum
F.P. of Iridium
P. pf Molybdenum
MOO
H
40MT
■ 3850 M, P, of Timgaten
Dun
Oxaage
Bjiebt
Orange
Whit?
Dasslme
1
I
M
ILLUSTRATION OF PRINCIPLES OF THERMOMETRY
19. Illustration of the Principles of Thermometry,
Absolute or Thermodynamic Scale
Gas Thermometers
(Primary standards)
27
Constant Volume
Constant Pressure
Gravimetric
Thermometers
(Compensated
Air thermometer)
'Secondary Thermometers
r — r T" ~i
Expansion "Resistance Thermoelectric Radiation
Thermometers Thermometers Thermometers Pyrometers
Bo oka R ecom men (led
1. Burgess and T.e Chatelier, The Measurement of High
Temperatures.
2. Ezer Griffiths, Methods of Measuring Temperature, (Griffin,
1947).
■. Hcrniing, Temperaturmessung.
4, A Dictionary of Applied Physics (Glazebrook), Vol. J, Article
on Thermometry.
5, J. A, Hall, Fundamentals of Thermometry, Institute of
Physics, London (1953).
6, J. A. Hall. Practical Thermometry, Institute of Physics,
London (3953).
Other References.
Temperature, its Measurement and Control in Science and
Industry (1941) , published by Reinhold Publishing Corpo
ration,, New York.
METHOD OK MIXTURES
24
CHAPTER II
CAL0R1METRY
1. Quantity of Heat. — It is a matter of common experience that
when a hot body is placed in contact with a cold one, the former
becomes colder and the latter warmer ; we say that a certain quantity .
of heat has passed from the hot body to die cold one. But a sim
experiment shows that when different, bodies are raised to the same
temperature and then allowed to exchange heat with a cold body the
final temperature is different. If we take equal quantities of water
in three different vessels at the same temperature and plunge equal
masses of aluminium, lead and copper previously heated to JOO'C into
diese vessels, one in each, the equilibrium temperature is highest for
aluminium and least for lead. This indicates that, of these three
metals, aluminium can yield the largest quantity of heat and lead
the le
For measuring quantities ol: heat we require a ' unit/ The
quantity of heat required to raise the temperature of 1 gram of
water through 1°C is called the 'calorie' which is also the thermal
unit tot measuring quantities of heat. The 15°C calorie is defined
as the quantity of heat, which would raise the temperature of oi
gram of C to 15.5°C and has been re ided hv
the lied Physics (1934) for
andard. In Britain the British thermal unit
tly employed* which represents the
qua heat required to raise 1 lb. of water through l°I\ 1
of any substance is defined as the number of calories
. u of the substance through 1°C. Tin's is
peaking not the same at all temperatures. Thus if a quantity
. the temperature of m grams of a substance from b
to 8% s f the mean specific heat of the substance, is given by
 m {B'~9) ] ; while if a quantity dQ raises the temperature by d&*
tin specific heat at the temperature $ is —^ ,
The thermal capacity or water equivalent of a particular body
he product, of its mass and specific heat.
2. Methods in Calorimetry.* — 'The following are the chief methods
iloyed in Calorimetryf : —
(1) Method of Mixtures.
(2) Method of Cooling.
* Sometimes the lb. calorie or cen%rade heat unit ("C. H. U.) or centigrade
thermal unit (C. Th, U.) is also used which represents the quantity of hi
ed bo raise 1 3b. of water through 1'C.
good account of these method * js given in Glazebroafe, A Dictionary of
Physics, Vol. I, article on "Catorimetry".
(»)
Methods based on Change of State or Latent Heat
Calorimetry,
(■1) Electrical Methods.
In die following pages we shall discuss these methods one by
der each of these we shall consider the various forms of
mental arrangement that have been adopted. Solids and liquids
will be considered first while gases will be taken up later in the
iter.
L METHOD OF MIXTURES
3. Theory of the Method. — Regnault* about the year 1840 made
,i careful study of the Method of Mixtures, and by 'care and skill
icd results of the highest accuracy. The principle of the m.
is to impart the quantity of heat to be measured to a certain mass
of water contained in a vessel of known thermal capacity and to
tire the rise of temperature produced. Thus, if a substance of
,72,., specific heat s t and initial temperature 6 Xf be plunged into
rams of water at temperature 0g* and if W be the thermal capa
of the calorimeter, the final temperature of the mixture, we
1 1 i : . i. • , by equating the heat lost by the ice to the heat gained
by the water and calorimeter,
lis gives the specific heat of the substance. Various correc
r, necessary for heat is lost by the system by
luction, convection and radiation. Thus lor 8 we must put
' e /S& is* the correction.
A. Radiation Correction, — In most experiments on calorir
the calculation of this loss of heat due to radiation is important.
ITic radiation correction may be accur
al Hy calculated with the help of
ton's Law of Cooling (Chap. XI) &
h states that for small differences
of temperature the heat loss due to
iation is proportional to the tempera
difference between the calorimeter
i iul the surroundings. To illustrate its
application kit AB (Fig. 1) denote the
rved rise of temperature during an
ent, RC the observed cool:
i, I of it. We have to calculate
rue rise in temperature*. Divide
the abscissa into n equal intervals 5.',.
:.i....St B by means of ordinate* P 1 M ii P 2 M 2 , . . .P„M S such that
* Henri Victor Regnault (18101878), born at AixJaChapclle, had to support
himself while young, He joined the Ecole Poly technique in Paris and later on
in [840 he was appointed Professor in that PoJyteehnique. He r.id many classic
. on heat.
AM.MW W«
■tf
fl
+ D
Fig. 1. — T] lustration of Radiation
CoiTU'/li: 'I.
30
CALORIMETRY
CHAP,
the small portions AI^ PJ* . .. P^P., may be treated as straight
ines. Let us measure temperatures from th<
ie temperature or the
surroundings. If 8 X , 2 . . . denote the mean temperatures during these
intervals, B t ', 0/ . , . the temperatures at the ends of these intervals
represented by P^, P 2 M 2 . . „ then the temperature diminution due
to radiation in the interval gr^ is kBtfh. If ${, 0*, . . . denote
the temperatures at die ends of these intervals had there been no
loss due to radiation, then
0"^
= 6,
t "= #,'  kB x U x
a "= 0/ f MjBtj,  ke.,bt 2 .
+ Af^&i ! 8 a/ 8 f .. .. dji„]
+ £(area of the curve ABM„A)
= V + k
'j:
e *//.
(i)
We can thus correct any temperature 0/
AP.M.A and k.
if we determine the area
An alternative method is to plot the upper curve from the lower
curve by increasing the ordinate M t P x to MiP/, M,P. to M«P,' etc
where M^P/ = ^ HP,'=V. etc. The highest ordinate "on the
curve (wo. D£) gives the true rise of temperature in the experiment
corrected for radiation. r
To determine h we have to observe the rale of cooling at anv
temperature. The curve BC {Fig, I) is obtained experimentally for
this purpose. From this ~ is calculated for any mean value of $.
is known.*
Another method called the adiabatic method is to eliminate the
teat bj continuously adjusting the temperature of the bath enclosing
me calorimeter to be always equal to the temperature of the caTori"
ureter itself.*
5. Specific Heat of Solids. — For finding the specific heat of solids
by this method the requisites are a calorimeter with an enclosure, a
thermometer and a heater. For work at ordinary temperatures the
calorimeter is made of thin copper, uickelplatcd and polished on the.
outside., so as to reduce radiation losses, it is supported on pointed
pieces of wood or by means of thread inside a larger doublewalled
vessel which has water maintained at a fixed temperature in the
,.;, t,A« fl rt nple b v fc ro V; ?h m ! thod sometimes adopted is to add to the observed
rise halt the coolwyr observed at the hi R hest temperature m a time equal to the
duration .of the experiment. This is based on the assumption that the average
excess of temperature of the calorimeter over the surroundings may be taken In
«Uw i tt i T^ S > hL T cc the co ° ]in « durin ff the experiment is half th
cooling at the final temperature.
* tt F Ja other methods £cc Glazebrook, A Dkiiottary of Applied Physics, Vol. 1
SPECIFIC HflAT OV LIQ1 '" ■
31
D
la space between the walls. The heater is a steapjacket in
,, i substance is heated by steam without becoming wet. An
oil bath i i be used. The transference and radiation errors must
i by suitable mechanical devices as in Rcgnault's classical
in ents.
For high temperatures the solid substance is heated in an electri
furnace. While in his work at high temperatures employed a furnace
h .. in •■ .■ platinum coil wound on its surface. The substance is sup
inside the furnace in a loop of platinum wire
up! is allowed to drop into the calorimeter by a
i i mechanical device. Change in temperature
is measured by a resistance thermometer. For work
m tow temperatures the substance is cooled down in
a quartz vacuumvessel surrounded by liquid air
before being dropped into the calorimeter.
Awbery and E/er Griffiths have determined the
irk heat of solids and molten liquids as well as
i latent heat by using an improved apparatus
i iid on the method  of mixtures. This is discussed
in Chap. V.
The use of water as calorimetric liquid has
eral drawbacks. Its range is small and specific
large so that the rise of temperature is small ;
lier there is considerable risk of some water being
1 1 i,i by evaporation. For these reasons several workers
have replaced it by a block of metal The copper
block calorimeter devised by Nernst, Lindernann and
Koref is exceedingly convenient for low temperatures.
It consists of a " heavy copper block X (Pig. 2)
.Hinted with Wood's metal to the inside of a Sewer
flask D. It is essentially a calorimeter based on the
method of mixtures in which copper replaces water
as the standard substance. The heated substance is
iped into die copper block through the glasstube
11 and the change in temperature of the latter is read
on thermocouples T, T, whose one end is inside the
copper block K and the other end in the block G.
The copper block on account of its good conductivity keeps the
temperature uniform. Jaeger and his coworkers have employed this
method to determine the specific heat of W, Vt, (X Rh, Ir, etc., to
about, lGOn^C with a high degree of accuracy.
6. Specific Heat of Liquids. — Specific heat of liquids which do
not react chemically with water or any other substance or known
specific heat may be obtained by direct mixture. For liquids which
react in this way Rcgnault used a different form of apparatus. The
liquid was not allowed to mix with water but was admitted when
desired into a vessel immersed in water. The liquid was first heated
•T
Ffc. 2.
Copper Block
Calorimeter,
CAl.ORIMETRY
[CHAP.
The specific heat could be calculated
and then forced into the vessel.
as before.*
Another cla*s ol experiments For measuring the specific heat, of
liquids involves the expenditure af some mechanical energy and
measurement of the consequent rise of temperature. To this class
belong die classical experiments of Rowland for determining the
ichamcal equivalent of heat. They will be described in detail in
Chapter 11L
2. METHOD OF COOLING
7. This method, perfected by Dulong and Petit, is found to be
most, convenient for liquids but unsuitable for solids owing to varia
tions of temperature within the latter. The method is based on the
assumption that when a body cools in a given enclosure on account
of radiation alone, die heat dQ emitted in the time dt is given bv
die relation
d(l=Affldt, . . . . ,(4)
1 depends upon the area and the radiating power of the Em
f{9) is an unknown function of $, the excess of the temperature of
tile body over that of the surroundings.
I! :' ; produce* a cooling; of the body through d$, we have
dQ = msdd,
iote the mass and die specific heat of the body
ng these two expressions for dQ we get
 made =3 Af(&) dt.
Or f .. m * a dO
Or t __ ms f* 1 ii8 _
" * )&*Mr ' '  '  I 5 )
e * is the time the body takes in cooling from X D to a 2.
Similarly for another substance to cool through the same" interval
Ml LcMl'J J _lcl <X Lll] C
' A' )B s f(d)' ' ( 6 )
* i ~: r i e * Che sutfa f e area and the radiating power of the
two bodies be rbx: same we have from (3) and (6)
ms m's'
t = r ■ ■  (7)
If masses m, m> of two liquids be contained successively in a
calorimeter o thermal capacity W and the calorimeter Upended
* For details see Preston, Theory of Hmt
u
MFTUOD OF MILTING ICE
,
inside a vessel kept at 0°C by immersion in melting; ice and then
.a lions of the rate of cooling taken, we have
W_ ■
t
*o
(S)
II' one liquid is water (s — 1) , the specific heat of the other is thus
determined from a knowledge of t, t', m t m'.
The method is sometimes employed for determining the specific
li it of liquids but is not capable of any great accuracy and is mainly
of historical interest.
3. METHODS BASED ON CHANGE OF STATE
8. These methods may be subdivided Into two : namely the
method of melting ice and the method of condensation of steam.
These methods were of real advantage in the last century when
accurate measurements of temperature were impossible, but with the
recent development of accurate thermometers and electrical heaters
they are now less in use, chiefly on account of their inherent defects.
The second method is, however, very convenient for determining the
specific heat of gases at constant volume and hence retains its
importance.
9. Method of Melting Ice.— In this method the heat given out
by a certain substance in cooling is imparted to ice and measured by
the amount of ice thereby melted. Thus ir M .grams of a substance
of speciQc heat a and initial temperature & are able to melt m grams
of ice when placed in contact with the latter, the specific heat is
given by the relation
MsB — m.L, . . . .
where L is the latent heat ol; fusion. The earliest forms of the appa
ratus as devised by Black, and by Lavoisier and Laplace were liable
to cause considerable error. An improved form of the calorimeter
was later devised by Eunsen* which will now be described.
10. Bunsen's Ice Calorimeter. — In this calorimeter, the water
produced by the melting of ice is not drained off but is allowed to
remain mixed with ice and the resulting change in volume is observed.
The calorimeter is illustrated in Fig :<. p. 34. The test tube A is fused
into the cylindrical gla^s bulb B which is provided with the glass
stem C. B is nearly filled with boiled airfree water and the remain
ing spare and the stem is filled with mercury. The stem terminates
an iron collar D containing mercury into which a graduated
tillary tube E is pushed so that mercury stands at a certain
• n in E.
In conductio , an experiment a stream of alcohol, cooled by a
ezing mixture, is first passed through the testtube A until a cap
♦Robert Wffliebxi Bi 1111899), born at Gottingea, studied at Gottingen,
Paris, Bcriin and Vienna. He r of Chemistry at Bresfau and
berg. His important researches arc on spectrum analysis Bunsen cell,
'. i; •:••• gas tntni :r and ice calorimeter.
3
34
CALORIMKTRY
[CHAP.
of ice F is formed round it in B. The whole instrument is then
kept immersed in pure ice at
0°C for several days till all the
water in B is frozen. It is then
ready, for use.
To calibrate the scale on E,
let a mass m of water at a tem
perature 0°C he poured into the
testtube. Some ice in B
melts and the resulting contrac
tion of mercury, say n divisions
on E, is observed. Then if a
recession of mercury by 1 divi
sion corresponds to q calories of
heat
rn$ = nq ; or q = mQ/n .,■ (10)
Next the substance un4er
i vivc:, ligation, previously heated
to a temperature &', is drooped
into some water at 0° contained
in the testtube A. Then if M,
Fig. 3^Bui Calorimeter. s d enote the mass and the spe
tanoe respectively, and v? the observed reo
oJ mercury thread in E, we have
Mse' = n'q,
or
m q "
Mnff
(11)
ill ilic heat of the substance. The specific heat,
of rare metals which can be had in small quantities can be readily
I by this method. The apparatus is, however, not capable of
rear accuracy. A fundamental objection to the use of the ice
calorimeter rests on the fact that a given specimen of water can
i: into ice of different densities,
11. Joly's Steam Calorimeter,— In the steam calorimeter devised
by Prof, Jolv in 1886 the heat, necessary to raise die temperature
of a substance from the ordinary tempera Lure to the temperature of
steam is measured by the amount of vapour condensed into water at
the same temperature. It consists of a thin metal enclosure A
(Fig. 4) , doublewalled and covered with cloth, which is placed
lath a sensitive balance. One pan of the balance is removed and
from this end of the beam hangs freely a wire w supporting a platinum
pan inside the enclosure The substance whose specific heat is required
is placed on this pan and weights added on the other pan till balance
is attained. The temperature of the enclosure is observed by means
of a thermometer inserted into the chamber, and in the meantime
tL]
jqly's steam calorimeter
35
steam is prepared in the boiler. It is then admitted suddenly into
the chamber through the wide opening O at the top and can escape
Fig. 4. — Job's Steam Calorimeter,
through the narrow exittube t at the bottom. Steam condenses on
the substance and the pan, and weights are added on the other pan
to maintain the equilibrium. When the pan ceases to increase in
weight the readings are noted and the temperature of the steam read
on a thermometer. During the final weighing the steam is all
to enter die chamber through a narrow escapetube so as not to disturb
the pan. The weight becomes practically constant in four or five
minutes though a very slow increase of about 1 milligrams per hour
may be observed due to radiation. The difference between the two
weighings gives the weight of steam condensed.
If W is the weight, of the substance, w the increase in weight
of the pan, $ t the initial temperature of the enclosure, $ s the tempera
ture of steam, k the thermal capacity of the pan, L the latent heat
of steam, s the required specific heat, then
k is determined from a preliminary experiment without any substance
on the pan and thus the specific heat of the substance is found.
For great accuracy •various precautions and corrections are neces
sary. Steam condenses on the suspending wire where it leaves the
chamber and then surface tension renders accurate weighing difficult.
A thin spiral of platinum wire in which a current Hows, usually
surrounds the suspending wire just above the opening, and is made
to glow so that the heat developed is just sufficient to prevent con
densation, A rapid introduction of steam is necessary in the early
s, for .steam also condenses on the pan due to radiation to the cold
air and the chamber, thus causing error. This is, of course, partially
balanced bv radiation from the steam to the substance later.
3G
CAL0RIMETRY
[< a ip.
Further w does noi accurate!}' represent the weight of si earn con
densed since the first weight is taken in air at 0^Q and the second
in steam at Ss°C. All the weighings must be reduced to vacuum
and then the increase in weight calculated. Specific heat of raw
substances ran be found by this method since small quantities ui
the substance are needed but a sensitive balance is indispen;
The specific heat of liquids and powders can be found by enclosing
them in glass or metal spheres whose thermal capacity is taken into
account. Cases can also be similarly enclosed, but then the modified
form of the apparatus — the differential calorimeter — is us<
12* The differential Steam Calorimeter, — In this form invented
by Prof. Joly in 1889, both the balance pans are made exactly
similar and of equal thermal capacity
and hang in the same steamchamber
(Fig. 5). The substance to be tested
is placed on one pan and the ex
cess of steam condensing on this pan
over that on the other pan is entirely
due to the substance. Thermal can. i
of the pans, radiation from them and all
other sources of error common to them
are eliminated, the substance bearing
only its own share of the error. The
chief use of this apparatus, hoy
consists in the determination of the spe
cific heat of gases at constant volume.
The pans are then replaced by two equal
hollow spheres of copper furnished with
"catchwaters" (shown in the figure),
One sphere is filled with the dried
S mental gas at any desired
sure while the other is empty. These
spheres are counterpoised by adding
necessary weights m which represent
the mass of the contained gas. St*
is admitted and condenses on the pans.
A larger amount of steam condenses on
the sphere containing the gas, the
excess, say w t giving" the amount of
steam required by the gas. Now the
specific heat, at constant volume c„ may
be calculated from the equation
'fy— e t )=wLt (13)
e $& 3 are the final and initial temperatures of the chamber.
Prof. Joly used copper spheres of diameter 6.7 cm. and weighing
pa. and employed gases at different pressures. Com tio
applied for the following : —
1. The expansion of the sphere due to increased temperature
the consequent, work done by t,lte gas in expanding to this volume.
:  itiHl St. 'run
Colorimeter.
ll.j
iiUiCTRl'CA I . METHODS
o7
2. The expansion of the sphere due to the increased pressure
mi the L;as at the higher temperature.
le thermal effect of this stretching of the material of which
the sphere is made.
4. The increased buoyancy of. the sphere due to its increased
tie as die higher temperature.
rhe unequal thermal capacities of the spheres.
6. The reduction of the weight of water condensed to its weight
acu
Dewar has devised calorimeters based on an analogous principle in
lie employed a liquefied gas as the calorimetric substance. Tiie
t. to be measured is applied to the liquefied gas whereby the liquid
i vaporates absorbing its latent heat and the volume F of gas thus
duced is measured. The heat communicated to the liquid is then
en by V pL where p is the density of trie vapour and /. the latent
at of vaporization of the substance. Using liquid oxygen and liquid
d ogen the apparatus can be adopted for very low temperatures.
1 1. the case of hydrogen 1 c.e. of vapour at N.T.P. corresponds to a
small quantity of heat (about 1/100 calorie). This method has
id for measuring the specific heat down to very low tempe
The experimental substance (solid or liquid) is first kept in
itant temperature bath (say 0°C) and then dropped into the
containing liquid oxygen or liquid hydrogen.
4, ELECTRICAL METHODS
13. The electrical method was first employed by Joule in his
npts to determine the mechanical equivalent of heat. The
trical methods at present available may be subdivided into two : —
(1) Method based on the observation of rise of tempera:
:'') Method employing the steadyflow electric calorimeter.
We shall first consider the application of these methods to liquids
mse historically the method was first applied to them.
14. Methods Based on the Rise of Temperature. — Following Joule
this method was adopted by many workers the chief among them
being Griffiths, Schuster and Gannon, W. R. Bousfield and W. E,
BouSfield. They employed Bliss method for determining tile mecha
lical equivalent of heat and found that it was capable of the highest
■ uracy. The same arrangements may be employed for finding the
he heat of liquids.
The principle of the method is to generate heat by passing a
renl through a conducting wire. If i is the current through the
wire of resistance R and E the potential difference across its ends, the
spent in a time t seconds is Eit ergs, provided K and i are
ed in electromagnetic units. If this raises the temperature
ol M grams of a substance by M°, the specific heat s of the substance
is given by the relation
EU—JMsM, (14)
ere / is the mechanical equivalent of heat (see Chap, J if). If E
38
CALORIMK'fltV
[chap.
is expressed in volts and i in amperes the energy spent is give
 anles (1 Joule = 10 7 ergs.).
Any two of the quantities E, i and R may be measured, Lhus
giving three methods. Griffiths, in his determination of the sp
neat of water, chose to measure E and R which is rather difficult
for R must be measured during the heating experiment, GiiffiLhs'
work is important since it first established the fact that the electrical
method can accurately give the value of J in absolute units, Schuster
and Gannon measured E and L
Jaeger and Steinwehr have applied this method Lo determine the
mechanical equivalent of heat and hence also the specific heat at
different temperatures. They employed a large mass of water (50
kg.) and consequently the thermal capacity, of the vessel was only
about 1% of that of the contained water. A section of their apparatus
is shown in Fig. 6. AA is the cylindrical copper calorimeter lying
on its side and properly insulated from the surrounding constant
temperature bath B. On the
upper side at O there is a hole
for the introduction of the
heating coil H t the resistance
thermometer and the shaft, t
which drives the stirrer SS. A
_nt. of about 10 amperes
was allowed to flow for six'
minutes through the const a n
tan heater H of S ohms resis
tance and the rise in tempe
rature was about 1.4 °C In
v experiments an accuracy
in 10,000 was aimed at
and hence the results are very
reliable,
IS. The Method of
Steadyflow Electric Calori
meter,— Great accuracy was
ained by Callendar and
Barnes by using the steady
How electric calorimeter shown in Fig, 7. A steady current of the
experimental liquid Sowing through the narrow glasstube /, about
2 mm, in diameter, is heated by an electric current flowing through
the central conductor of platinum. The steady difference of tempera
ture SB between the inflowing and outflowing water is measured by
a pair of platinum thermometers Ft, Pt at each end connected differ
entially in the opposite arms of a bridge of Callendar and Griffiths'
type, 'The bulb of each thermometer is surrounded by a thick copper
tube of negligible resistance attached to the central conductor. This
on account of its good conductivity keeps the whole bulb at the
temperature of the adjacent water, and due to its Ion resistance
prevents the generation of any appreciable amount of heat by the
Fig, 6. — A Section of jaeger and
Steinwehr' 5 Calorimeter.
STEADYFiLaW ELECTRIC CALORIMETER OV
current near the thermometer. The leads L, L and P., P are attached
to this tube of copper, die former for introducing the heating current
and the latter for measuring the potential difference across ihe central
uctor in terms of a standard cell by means of an accurately
Fig. 7.— Steadyflow Electric Calorimeter.
calibrated potentiometer. The potentiometer also serves to measure
the healing current i by measuring the potential difference across
a standard resistance included in the same circuit. In order to
diminish the external loss of heat the flow tube is enclosed in a herme
tically sealed glass vacuum jacket surrounded by a constant tempe
rature bath. Neglecting small corrections the general equation is
Eit=JMs (dsdj+jkt, .... (15)
employing the same notation as before, where h denotes the heat
loss per second on account of radiation, and & if &<, the temperatures
of inflowing and outflowing water. The time of flow t in these experi
ments was about 20 minutes and was recorded automatically on an
electric chronograph reading to 0.0 1 sec. The mass of water M was
measured by collecting the outflowing water and was about 500 gm.
The difference in temperature Q*—9t was from 8° to 10 D C and was
accurately read to 001 n C. The heat loss h was very small and
regular, and was determined and eliminated by suitably adjusting the
i trie current so as to secure the same rise of temperature for
different rates of flow of the liquid. Thus for two rates of flow we
have
E^t=JM v t{8 t d x ) f Jht,
•' J iM x M t ) [OtW
Since the temperatures at every point of the apparatus are the same
in both experiments the heat loss h must also be the same. The
i ecific heat s thus determined is the average specific heat for the
interval W and may be taken as the specific heat at the middle of
the interval.
The great advantage of this steadyflow electric method is that
no correction is necessary for die thermal capacity of the calorimeter
40
CALORIMri'RY
[CHAP.
since there is no change of temperature in an) part of the instrument
Care must, however, be taken to secure perfeci steadiness, as it is
practically impossible to correct for unsteady conditions. Further,
since all condition?, are Steady, the observations cart he taken with
tiie highest degree of accuracy. There is no question of thermo
metric lag. It is essential, however, that the current of water be
thoroughly mixed otherwise temperature over a crosssection of the
tube will' not be uniform. This is secured by having! the central
Wictor in the form of a spiral instead of a straight wire,
Callcndar and Barnes used this method to find the specific heat
of water at various temperatures. Their results are discussed in the
next, section Callendar found the specific heat of mercury by this
method. The central conducting wire was dispensed with, die flowing
mercury itself serving as the conductor. Griffiths employed this
method to determine the specific heat of aniline over the range i
to 50°C.
16. Specific Heat of Water. — In ordinary calorimtftric experi
ments the specific heat of water is assumed constant, at all tempera
tures and equal to unity. Accurate investigations of the last section,
show that it varies with temperature. The first accurate experiments
in this connection were those of Rowland in connection with his deter
mination of the mi' ical equiv: leni " : heat (Chap. 111). He argued
e specific heat of water at all temperatures were constant
this mechanical equivalent must come out a constant quantity even'
used water at different temperatures. The variation in the value
irk heat.
?■ at different le\ wres.
mp.
Callendar
Jaeger and
Osborne
Specific heat
arnes
Stemi
Stimson &
5n int. Joules
(O, S, & G.)
1.0093
(1.005)
1 .0076
1 4.2169
.1
1.00f7
1.0029
1.0O39
1.2014
10
1.0019
1.0013
1.0015
4.1914
15
1 .0000
1.0000
oooo
11850
20
0.9988
0.9990
9991
4.1811
25
0.9980
0.9963
0.9fl
4.1788 '
30
0.997ti
0.9979
9982
4.1777
35
0.9973
0.9978
0.9982
4.17.
40
0.9973
0.99R1
0.9983
4.1778
45
0.9975
0.9987
0.9985
4.1787
50
0.9978
0,9996
0.9988
4.1799
60
0.9987
0.9997
4.1836
70
1.0000
1.0009
4.1888
80
LOO 17
1.0025
4.1956
90
1.0036
1.0046
4.2043
100
L0057
1.0072
4.2152
iu
NERNST VACUUM CALORIMETER
41
The oilier accurate experiments on the subject are those
sndar and Barnes (sec. 15) and of Jaeger and . Steinwehr (sec 14).
Both of them determined accurately the specific heat of water at
us temperatures, Their values are given in Table 1 together
with the value;, obtained recently by Osborne, Stimson and Ginnings
at; the National Bureau of Standards, Washington. In column 5 the
specific heat is expressed in international Joules* per gram per °C.
40 s ST
TSUPERAI b*RE K
II!..'
Ft 8.— Specific Heat Curve far Water.
he results of all these three investigations arc plotted in Fig. 8.
It will .he seen thai the values obtained by Callendar and Barnes lie
somewhat wide of the others and appear to be less reliable chiefly on
account of the uncertainty in the values of the electrical units
employed. From these curves it is evident that water has a minimum
ific heat at about 34 n C. It is on account of this variation that on
IB the calorie was defined with respect to 15 C C.
Specific Heat of Solids
17. Rise of Temperature Method.— The electrical method was
first applied to solids by Gaede in 1902. E. H. Griffiths and E.
Griffiths determined the specific heat of many metals over the range
 160° to  100°C. The substance was used in tire form of a calori
meter and was first cooled below the desired temperature. Electrical
energy was utilised in heating the calorimeter and the temperature
led by a resistance: thermometer. The calorimeter was enclosed
in a constant temperature bath whose temperature was kept constant
to 1 /100th of a degree. Correction was applied for the heat lost by
radiation,
»yl8. Nernst Vaeuam Calorimeter.— A different form of the appara
tus, known as the vacuum calorimeter, was used by Nernst and
Lindemann for measuring die specific heat at very, low temperatures.
This differed from Gaede's form essentially in having the calorimeter
suspended in vacuum. The results achieved with its aid are of great
theoretical importance and hence their apparatus will be considered
* 1 /Int. Jodie = 1.00041 X 10 7 ergs.
CALORJMETRY
CHA '.
in some detail, T?or good conducting solids such as metals the
calorimeter shown in Fig, 9(a) was used. The substance whose
specific heat is to be determined is shaped into a cylinder G, having
a cylindrical hole drilled almost through its entire length, and a closely
fitting plug P made for it from the same material. The substance
here acts as its own calorimeter. The plug is wound over with a
p
c
fa) ( b) (C5
Fig. 9.— Nernst Vacuum Calorimeter,
spiral wire of purest platinum (shown dotted in the figure) which is
insulated from it by means of thin paraffined paper, and finally liquid
paraffin is poured over it. The upper part of the plug is somewhat
thicker than the lower part, thus a good thermal contact is obtained.
The calorimeter K thus constructed Is suspended inside a pearshaped
glass bulb [shown at (b)] which can be filled with any gas or evacuated.
The whole can be surrounded by suitable low temperature batlis such
as liquid air or liquid hydrogen. The platinum spiral, which serves
both as electric heater and resistance thermometer, is connected in
scries with the battery B, resistance r and a precision ammeter A,
the voltmeter V indicating the potential difference across the spiral.
In order to bring the calorimeter to the desired tempeiature of
experiment., hydrogen which is a good conductor of heat was first
admitted into the pearshaped vessel and the latter surrounded In a
suitable bath. Next the vessel was completely evacuated so that the
heat losses from conduction and radiation were almost entirely eli
minated. Tn addition it was surrounded by liquid air or liquid
hydrogen.
it.
RESULTS or EARLY EXPERIMENTS
experiment a current was allowed to flow through
xnds and the voltage across it was adjusted to be
denote
To carry out an
the heater for I seconds and the voltage
constant by varying the resistance r. If Rf, Ri and i fs
the final and initial values of the resistance of the heater and the
enl through it respectively and E the constant potential difference.
Thus an observation of it, i t and E gives R f and R, » and from a
previous determination of the resistance oE the platinum spiral at
various temperatures the rise in lemperature 80 can be found. The
energy supplied electrically is Eit where i is the average value of the
current. Now if M is the mass of the substance forming the calori
meter, s its specific heal, we have,
Eit = JMsM f h. . . . . (15)
This gives the specific beat at a single temperature since BO is usually
1° or 2°. The heat capacity of the paper and paraffin can be found
and eliminated by taking different amounts of the substance and at
the same time arranging that the temperature rise is the same The
heat loss h is very small and is determined and accounted for I 
observing the rare of cooling before and alter the experiment.
For nonconducting solids' the calorimeter shown in Fig. 9 (c) was
employed. The heating w T ire was wound over a cylindrical silver
vessel D and the whole covered with silver foil to diminish heat loss.
This foil was soldered at the bottom of the cylinder as indicated. The
solid whose specific heat is required was placed inside the silver
cylinder and the latter closed with the lid. The silver on account of
its high conductivity keeps the temperature, uniform and this is
further secured by filling the cylinder with air through the tube in
the lid. The tube is then closed with a drop of solder so thai it
mav be gastight. It is absolutely necessary that air should be
sent inside the vessel to facilitate equalisation of tern pern lure
throughout, the experimental substance. Liquids and gases can be
similarly admitted into the cylinder and their specific heat determined.
19. Results of Early Experiments.— In 1819 Dulong* and Petit
from their investigations concluded that the product of atomic, weight
and specific heat 1 xf/as constant for many substances, or in other words,
ims of all substances have the same capacity for heat. Regnault
from his own researches found that for ordinary substances the mean
value of the constant was 6.38 with extremes of 6.76 and 5.7. A
more accurate value of the constant can be obtained from the kinetic
theory (Chap. III). The atomic heat at constant volume is shjwn
there' to be equal to 31? = 5.955. According to Richarz, the value
::T the ratio Cp/c e for many substances lies between 1.01 and 1.04,
hence the atomic heat at constant pressure, the quantity commonly
determined should lie between 6,01 and 6.19. This law T is of great
* Pierre Louis Dulongr (17851838), a distinguished French scientist who lost
an eye and a finger owing to the explosion of some nitrogen chloride which he
discovered.
CAI.ORIMETRV
CHAP.
aiming atomic weights. Tn illustration of the law table 4
 is adde .
Ta ble 2 — Illustration of Dnlong and Peti
Element
am
M; '_
AJuminiitiii
Iron
led
Copper
Zinc
Silver
lium
Antimony
LI
Atomic Weight
Mean specific
.
(1)
IlC Hi
heal
: (2)
23*00
0.307
7.06
L32
0.247
li.nii
27.1
0.2175
5,83
55
0.110
6.14
.68
0,1092
6.41
63.57
0.0930
6&,37
0.0939
107.0
0.0559
6.03
11.
0.05
6.26
118.7
0.0556
121.8
0.05
6,10
[95
0.0318
6.21
197.2
0.031
8.10
0.0310
209
0.0299
ies= 6.24
,n mn enunciate aw concerning molecular
heat and
onstant The
 ; ll "' : " ' mounds to another In
illustration of the tbkf 3 is added.
—Moteo at of Oxides.
Compound
i tfic heat
Cr s O a
Bi a O,
Moleculai . Secular
dit heat
(2)
0.1700
0.1796
0.1277
0.0901
0.0605
159.8
192.0
197,8
287.fi
27.2
27. J
.
25.9
Mean value = 26.8
• ! :en f rom Vol S n 193
' lake!1 : Experimental ! ' g p , 200,
■".
TWO St'I CI! [CI
Limann's law can be considered as a particular case oE die
following law : 1 i ular heat of a compound may be considered
is the the atomic heats ol Its constituents. Thus if a com
pound has the composition A a "&i,G c T)j its molecular heat C t is given
n the relation
C p =aC,, , D . . . . (17)
B, C, D stand for the different types of atoms composing
the compound and C pjLt C p1i , etc., their atomic heats given by
mlong and Petit's law. The law is ol much use in evaluating the
nolecular he. [lain substances,
20. Variation of Specific Heat with Temperature. — The Specific heal
determined by the foregoing methods is not found to be a constant
juantity. For solids and liquids the effect of pressure on specific
i rather small. The effect of temperature is however very
d. Increase of temperature invariably increases the specific
feat, while the decrease of temperature lowers it. In fact the atomic
lea solid almost vanishes at the absolute zero and gradually
increases with rise of temperature reaching asymptotically the Dulong
bid Petit's value of 3JS at a sufficiently "high temperature which is
Iiflerent for different substances. This variation for silver is illus
d in Table 4.
—A torn ic heat of silver at different temperatures.
Temp, in c Iv
1.35
S
10
20
36.16
0. i:'V
0.00509
0.0475
0.399
1.694
Temp, in °K. Atomic heat
55.88
74.56
103.1 I
205,3
3.186
:.u;'
5.373
5.605
The asymptotic value oF $R can be accounted 1 for by the Kinetic
Theory (Chap. Ill) . The variation with temperature has been
successfully explained by the quantum theory of specific heats and
in particular by the Debye's theory of specific heat which is how
beyond the scope of this book.
21. Two Specific Heats of a Gas. — The specific heat of a gas.
as of solids and liquids,, may be defined as the ratio of die heat
bed to the rise in temperature, taking a unil mass of the gas.
A little consideration will show thai Emition requires to be
1 'ne a quantity of gas to be suddenly compressed.
The ' ii : oJ the gas will be found to rise, though no heat has
been added. The rati •' added/increase in tempera' v.
ic heat, vanishes. Again let this com] air expand
; ]]'.. ; :\ roolrng v. ;ikc place. Tins is just prevented bv
applying some heat to the gas. In this en.se the ratio, heat ad
415
CALORIMETRY
CHAP.
EXPERIMENTS OF GAYLUSSAC & JOULE
47
change in temperature becomes infinite. Thus we see that the original
definition gives an infinite range o£ values for the specific m
Hence external conditions are of paramount importance in deter
mining the specific heat of gases. It has become customary to
ak of two specific heats o£ a gas : the specific heat at constant
volume denoted by C, and the specific heai at constant pressure
denoted by tip. In the former process the gas is maintained at
constant volume so that the whole heat applied g es to increase the
internal energy of the gas. In the latter case die gas is allowed to
expand against a constant pressure and in so doing it does external
work. This work is obtained by using up part of the heat energy
applied ^ to the gas. Hence the specific heat at constant pressure is
necessarily greater than the specific heat at constant volume by an
amount which is simply equal to the thermal equivalent of' the/
work done by the gas.
Let us assume that the gas is perfect, i.e., its molecules exert no
influence on one another. This is approximately true for the permanent
gases as Joule's experiment (Sec. 22) shows. Hence in this expansion
no internal work against molecular attractions is done by the gas and
the excess of heat supplied in the second case is simply the thermal
equivalent of the external work. Consider the gas enclosed in a
vessel of any shape and suppose the walls of the vessel can expand
wards. Lei &A denote an element of area of the walls and Bx the
nice traversed hy, it measured along its outward drawn normal.
the work done by the gas in this expansion against a cons
' pressure :.j^; and for die expansion of the enth
the work is equal to % p&AM = pl&AJIx — p&V where
lie volume of the gas. Suppose a gramme
ping a volume V t at temperature T°K and
pressure to a volume V, at temperature (T  1) Cl K the
og constant Let Mc p = C p , Mc p = C, where M h
the molecular weight of die gas. C* and C, may be called trram
22.
GayLussac first car
internal
■nol. molar specific heats,
expanding from V x to l% is
The work done by the gas in
P(V S  V t ) ^ R [ (T +]}T]=R J . . (18)
y the gas laws. The thermal equivalent* of this work is T(C* — C )
■ . . (19)
and C. measured in
Hence
JfC^C,) =Rf,
Or, if either R is expressed in calories, or C*
. : ■•■■ we get
C P C S =R.
This relation was first deduced by R. Mayer.
if] oS^mS^. FirHt Taw ° f T^^anfe which will be diseased
tThia is true for a perfect sas only. For real gases it cau be shown that
Cp ~ CB = T \dT) L ,\ arJpWhere the differentials
I be evaluated from the
Experiments of GayLussac;!: and Joule.
nc« out experiments to determine whether a gas does any internal
in expanding. He allowed gas contained in a vessel at a high
o expand into an evacuated vessel and observed the fall of
[en perature in one and the rise of temperature in the other vessel.
is called free expansion or Joule expansion. In this the gas as
whole does no external work while its volume increases, and the
work done will be that against molecular attractions which is
nal work. Somewhat later Joule employed a similar
iratus but he immersed the vessel in calorimeters. His apparatus
actual equation o£ state (Chapter IV) for the gas.
Joule's experiment with Joulc:^ experiment with
one calorimeter, two calorimeters.
Fig. 10 Fig. 11
indicated in Figs. 10 and 11. The two vessels A and B corn
" i a ted with each other through a tube furnished with a stopcock
A was filled with dry air at 22 atmospheres while B was exhausted.
prat the whole apparatus was placed in a single calorimeter (Fig. 10)
md the stopcock C opened. No change in temperature of die water
fas observed showing that no internal work against molecular attrnc
done I the gas in expanding. To investigate the point
iriher, the parts A, B/C were placed in separate vessels (Fig. II;,
attaining water whose temperature could he read by sensitive
lermometers. On opening the stopcock C the air expanded into B
id the temperature of the vessel surrounding A fell while the tern
>eratures of B and C rose. It was found that the heat lost by A
/as exactly equal to the sum of the heats gained by B and C,
jus the total change in the internal energy of the gas during expansion
zero. *.e\,\l— J — where U is the internal energy and P
ic volume. This is called Joule's law or Mayer's hypothesis. This
hows clearly that no internal work is done by a gas in expanding.
[oule's law holds only for the perfect gas to winch, the permanent
jases of Nature like helium, hydrogen, etc. approximate. For further
J Louis Joseph GayLussac (17781850) was a distinguished French scienti
fho investigated on the expansion of gases. He was interested in aviation and
1804 made a balloon ascent for the purpose of making experiments,. He was
a peer of France.
CALORiM '
[CHAP,
iSUKEMENT OP C*
49
discussion see Chap, \ I, sec. lf> and Chap. X.. sec 3. As a n
of fact, a slight fall in temperature should be observable in Ji
experiment with one calorimeter but on account of the I.
capacity of the calorimeter it escaped detection,
23. Adiabatic Transformations. — When the pressure and volume
of a substance change but no heat is allowed to enter or leave it,
the transformations are said to be adiabatic (a = not, dia = thro
bates = heat, i.e» 3 heat not passing through) • In an isd hermal i h
thi temperature is kept constam by adding heat to or taking it away
from the substance. Consider an amount o[ heat 8Q applied to a
perfect gas. This is spent in raising the temperature of the gas and
in doing external work. H we consider i jramnn olecule of the
gas, the former is equal to C r d'f and the latter equal ro pdV/J, both
in calories. Hei,.
MZ = C,dT + pdV/J,
or, if Sf) and C., are measured in ergs,
= C,dT + pdF. ...
This equation combined with pV = RTyn\l give the solution oi
all problems on perfect gases,* In an adiabatic transformation
80 = 0. Therei
C B dT r pdV = i
in order to find a relation between p and V we must eliminate
M (21) I jas equation pV = RT. DifTerentii
Vdp = RdT i
dT from (22) in (21) , we get
and re pi ii by C p C„.
C,Vdp + C p pdV = 0,
•'ting C p jC 9 by y we obtain
dV
i>«
which on integration yields
log p + y log V = constant,
or — constant,
is the adiabatic relation between p and V for a perfeel gas.
find the adiabatic relation between 7" and V or bet reen /'and
p we must respectively eliminate p or P betwei n ind
the gas equal
jpvi__ constant (24)
■ — constat 1 1 i
In ■"." bRT, and CVC I substitttl
equation oi state mid the true value oi C,C„ We
ion between />, F and 7 for real i^aes.
, ._Drv air enclosed at 25 e C and ai atmospheric pressure
,, 1 ,.] to half its volume. Find (a) the resulting
the resulting pressure. Assume y = 1.40.
!4) T 1 = T(7/7 1 ) 1 (273+25) (2)^ ~393«K.
: 'I \V x )vpV^ ut =264 atm.
24. Experimental Methods*.— Let us now consider the expej
ods oi Ending the specific heat of gases. Since for
ises C„ C B — R, a knowledge of one of the specific heats
other. Again, if we dejpmine y, i.e., C I ie above
and C B . Hence the experimental methods
ivided into three classes :
(a) The measurement oh (.',,
(b) The measurement oE C,,
(c) The determination of y.
: MEASUREMENT OF C p
25. The specific heat at constant pressure has been found either
by the Method o£ Mixtures or by the ConstantFlo thod. The
principles of these methods have already been explained.
26. Method of Mixtures— Regnault's Apparatus. The method was
first applied to gases by Lavoisier and Laplace. Improvements were
GE2T
I •!;■ imauh's Apparatus for CV.
'■ The reader will find a very good account of these meQwxLg in Partingti n
Heai oj Ga
50
CAUHUMi
CHAP.
made later by Delatoche and Berard, Haycroft and Regnautt.
Regnault with his groat, experimental skill obtained results of high
acy. His apparatus is indicated in Fiu;. 12. Pure dry gas was
compressed in Lhe reservoir A which was immersed in a thermostat.
Mi. reservoir was provided with a manometer (not shown). Gas
could be allowed to flow through the stopcock V at a uniform rate.
This was effected by continuously adji Ele stopcock V (shown
■ rely) so that the pressure indicated jry the manomel r R was
constant, The gas then flowed through a spiral S [n : in a
hoi nilbath and then into the calorimeter C fir:: ; into
lhe air. The gas acquired the temperature T oj rath and raised
the temperature of the calorimeter, say, from (^ to i),.
Tf m is the mass of the gas that flows into the calorimeter, r^ its
specific heat, is given by
(26)
mo (r A t*)»(V
w is the thermal capacity of the calorimeter and it* contents.
I 1 1 .: : mass m of the gas was determined by Regnault as follows; —
f€>r any pressure p he assumed that the weig lined
in the reservoir at temperature was given bv the relation
W(l + a&) =Ap+.r: C
B t C were determined, from leliminary experi
rresponding to the observed
dent is given
\p> f Bp
of the gas iras found.*
i>i gain <■'•
and radiation which have to be found out
.tare of the calorimeter before and
the experiment, and taking the avea
27. Other Experiments, The experiments of Wiedemann v
similar to those of Regnault. Lussana devised a high
ire apparatus in which the same amount of gas enclosed at
a high pressure can he repeatedly healed and passed through the
The principle employed is the same as in Regnault's
iment. This apparatus can be used to find the specific heal
o£ ery high pressure and possesses the advantage that the
gas is not wasted.
For determining lhe specific heat at high temper;) he experi
ment of Holborn and Henningmay he mentioned. Employing .u liable
floe trie heaiers, resistance thermometers and welldesigned calori
* If the i . ; relation po = RT/M (p. 5), where v i* the specific volume
inn: M its molecular weij assumed to hold, a simpler expression
For W ijned. Let V denote the volume of the reservoir
1 a'szl/v, the density of the gas. Then
W— V P = 3g£, and IYW'=™ (pf),
where p is cxprtHsed in dynes and R in ergs.
'RT
MEASUREMENT OF C,
51
able to find the specific heats of nitrogen, carbon
i team up to 1400°C. The calorimeter corrections are
uncertain. The same method has been used by Nernst
the specific heat of ammonia up to 600 °C.
2ft. Constantflow Method. For finding tire variation of specific
ii i nperature the constantflow method is most suitable, and
used by Swan. The most recent form of the apparatus is
thai used I !•. ill I'.' el and He use in finding the specific heat down to
ow temperature and is in principle similar to that shown in
1 to rider test., previously brought to a steady temperature
ugh a suitable bath, flows through the calorimeter in
i. Inside the calorimeter it is heau'd electrically by
in' roil of constantan and thus the energy supplied can be
• I i M iv, of the incoming and outgoing gases
ranee thermometers. The specific heat, can be
v an equation similar to (15).
li at low 1 mperatures the gas was initially passed through
i i rature bath in which the calorimeter was also immersed.
'hum and other rare Scheel and lleuse modified
o as to employ a closed circuit These experimenters
1 icasurements on various gases in the range 0° to  180 o C.
Foi measurement of specific heat at high pressure the constant"
od has been employed by Holborn and Jakob and gives
ih able results.
(b) MI'J HODS BASED ON THE MEASUREMENT OF C v
.Steam Calorimeter. The direct determination of C. is best
ol Joly's steam calorimeter (sec. 12). The method
th< periment and the necessary details will be found
on.
1J  1 ' i ■ • c l i, Mm \
32
CAIORIMKTRY
[CHAP.
30. Explosion Method* Following the work of Bunsen, Vieille
and others, Pier improved the explosion method and devised the
modern explosion bomb indicated in Fig. 13. Immersed in the water
bath B is a steelbomb A which has a side tube M through which the.
bomb can be evacuated and various gases introduced at the desired
partial pressures. S is a corrugated steel membrane closing an open
ing in the bomb and carrying a mirror S. Light reflected from the
mirror (alls on the photographic film F which revolves on a drum 3y
applying various known static pressures and noting the deflection of
the light spot, the pressure attained in any experiment can he found
simply from the record on the film, Any explosion mixture, say,
a mixture of hydrogen and oxygen together with tlit; inert ^as whose
specific heat is required, is introduced in the bomb, L>y 'inert' is
meant any gas which will not take part in the reaction either from
want of chemical affinity or due to its presence in excess. The
partial pressures of the various gases are known. The explosion is
started by means of electric sparks and the final pressure reached
is observed. This takes about .01 sec.
The calculation may be easily made. If the vessel were allowed
to cool to the initial temperature T 2 (absolute) , suppose it. would
record the pressure pn. The maximum temperature T, reached during
the explosion is calculated from the value of p x , for, from^the gas
, since the volume remains constant,
r[
T — Pit
Or ii ite the initial pressure and e. the ratio of the final to tb
1 number of molecules (owing to the explosion the total m i
of li
= e Pi [ '" volume and temperature are me]
■r —P* t 7"
(28)
where P is the ratio of explosion pressure to initial pressure. The
relation connecting the specific heats and the heat of reaction is
mih (TxTz) [mC ar \nC H ] f . . . (29)
where m is the number of grammemolecules of the reaction products ;
n the number of grammemolecules of the inert gas; C vr} C re ,resper
their mean molar specific heats over the range (T*  7Y) , and Q 2
the heat of reaction for the explosion mixture at T a °. C> £ is generally
known from thermochemical data. We can determine C pr by explod
ing either with different amounts of a gas whose variation of specific
heat with temperature is known, or with different quantities of argon,
a substance whose specific heat is constant. Then the same reacting
gases may be exploded with any inert gas, and knowing C v r , we can
find C Di for the inert gas.
The method is very suitable for measurements of specific heat
at high temperatures and has been used to about 3000°C but suffers
from the disadvantage that directly it gives only the average values
H I
DETERMINATION OF y
58
wide range and not the specific heat at any temperature.
iluv, corrections are necessary for loss of heat, effects of dissoeia
n, incomplete combustion, etc. For argon. Pier found that the
heat does not vary with temperature.
3L list's vacuumcalorimeter method is employed to find
ii low temperatures. The gas is enclosed in the calorimeter
j. Eucken in this way found the specific heat of hydrogen
. :>°K and obtained interesting results.
METHODS BASED ON DETERMINATION OF y
32. As already pointed out this is an indirect method of finding
the s iecil c heats of gases. Though indirect it is capable of the highest
v so that the modern accepted values of specific heats are
on the values of y thus obtained.
The methods for measuring y, the ratio of die two specific heats,
ii !'• ie classified under two heads : (I) those depending on the
■<ic expansion or compression of a gas, (2) those depending on
the ty of sound in the gas. We shall first consider the former,
(1) Adiabatic Expansion Method
33 Experiments of Clement and Desoraws*— Clement and Desormes
>: the first to find y by the adiab; tic expansion method. Their
.inal apparatus has been considerably improved and is indicated
in Fig. II, A large flask A of about 28 litres capacity is closed
I • a stopcock M about 1.4 cm. in diameter. Tlie flask is
iiumected to the manometer P t P,.[ by means of a sidetube and is
with cotton wool to avoid loss of heat. First the flask is
tialJy evacuated and the pressure p t recorded by the manometer, is
observed. The stopcock
M is then opened
and quickly closed. Air
ics into the flask till
i he pressure inside and
ide becomes equal,
i 1 i ■: ■ process is adiabatic
die loss of heat in
the short interval for
which the stopcock M is
• ii may be neglee
The temperature of the
air in the flask rises on
unt of the inrush of*
external air and the
are becomes atmos
pheric. The flask is next
allowed to cool to the
imperature of the sur
• 'i idings when the water
in I he :.■ On ter rises
and finally indicates the '
Fig. 14. Clement and Desormes' appai
 1 ™
54
GAL.OKIMETRY
 GfiAPi
Let the atmospheric pressure be p A and the specific volumes of air
at the pressures, p t p Ai p f , be respectively v it v A , i>/. The first: process
is adiabatic and hence we have, assuming the gas lo be perfect,
p z V'p.4l<A v TO
Since the final temperature is the same as the initial, we have,
considering 1 gram of the gas,
SpFi^PfVf :
Again, v A v ; . (32)
amse the volume of tile manometer tube is negligible compared with
that of the flask and hence there will s no appreciable .. auge in the
specific volume of the gas due of liquid in the manometer.
Combining (30), {81), and (32), we have.
Pa \pfh
yJSiCJEiiL.
log pi iogp f
II, as usual j the changes in pressure arc small,
' PP, f
From th irements of Clement and Disomies, Laplace deduced
lu< of y to be IM
In this experiment there is a source We have
1 that the pressure inside has become atmospheric when the
stopcock M is closed Actually, however, osdllatio on
'Lint of the kinetic energy more air first rushes in than would make
pressure just atmospheric, and hence the pressure, inside becomes
greater than p A . Next some air rushes out till the pressure inside «
is less than p A and so on. After several such overshooting? the
pressure p A h attained. This takes considerable time and, as a matter
of fact, this toandfro motion has not subsided when t,he stopcock
is closed. It must be closed at the instant when during an oscillation
the pressure just becomes atmospheric. This is very difficult to secure
and nence later investigators tried to avoid it by measuring the change
in temperature resulting from adiabatic expansion. The scopcoi
has not to be closed in this case. We shall consider shortly the experi
ments of Lummer and Pringsheim and of Partington and Shilling
baged on tills principle.
We have assumed above that the incoming air has the same
temperature as the air in the flask initially. To avoid correction in
v it is not so, it Is better to start with compressed gas in the flask.
when a Utube manometer must be used in place of t^P^. Further
care must be taken to use perfectly dry air lor y is appreciably
different for moist air. Consequently sulphuric acid is generally used
as the liquid in the manometer.
54. Experiments of Joule, Lummer and Fringsheun, and Partington.' —
Joule was the first to study the change in temperature by
EXPERIMENTS OF "PARTINGTON
Hllnbatic expansion or compression. Various investigators later
i mi>!o\cd this method to determine y . Air was compressed m a vessel
,1 it s temperature and pressure observed. It was then allowed to
i In pressure and temperature before expansion, p 2i
quantities after the expansion, we have, from (25)
.[ suddenly to atmospheric pressure and the change in tempera
noted. Trie calculations can be easily made. If p tl 7\ denote
T» the same
lv
n
(ly) log
! 
"
7",
8^8
Pa £ 1
tog Pi 1°$ Pi
" Y (log Pi  tog Pi)  C lo S T i  J °g r a)
y can be calculated,
Lummer and Pringsheim made considerable improvements m the
ratus for determining y by this method. They employed a 90litre
here and measured the change in temperature by the change
topper sph...
n i. sistance of a thin bolometer wire hanging atthe centre oJ this
A Thomson galvanometer having a period ot 4 sec was
Certain errors arc, however, inherent, in
. . null
instrument.
1 1 ir apparatus
ill llllll .1 LLia. .
In order to eliminate these errors Partington has further improved
pparatus. He used a large expansion vessel (130 litres capacity)
: bolometer of very thin platinum wire (.001 to .002 mm. m dta
SfFI
T
PJ 5 ft
\ j:
a • \ii
iftvwvvv—
iv ■ .■•.. • 
i
/
Fig. 15.— Partington's Appafettis
i witli compensating leads ; thus there was no lag. Further an
♦This is true for a perfect gas only. For real gases it requires mmiifi cation.
V
56
CALORUI'
CHAP.,
Einthoven string galvunor pable of recording temperature in
.01 sec. is used so that a detailed record of changes in tempera t Lire
of the gas during and after expansion is obtained. His apparatus is
indicaLed in Fig, 15, The vessel A is provided with the expansion
valve C which can be manipulated by means of the spring P and
whose size can also be varied, A is connected to the sulphuric acid
vi mometer M, the mercury manometer m, and the drying tubes F.
Thus carefully purified air enters A. Further the vessel A is kpt
immersed in a waterbath which is kept stirred by S. B is the bolo
meter wire (shown separately, in the figure^ and is connected In one
arm of a Wheatstone bridge. G is the string galvanometer.
The initial temperature was read on a car standardised
mercury thermometer T immersed in the bath and was given correct
to .01°. Then the resistance in one arm of the Wheatstone bridge
lowered to give some deflection in the galvanometer. It was so
arranged by trials that immediately alter expansion this deflection
was reduced to zero. After the expansion experiment some ice was
continuously added to the bath to keep its temperature constant and
equal to than immediately after expansion. This was ascertained by
keeping the galvanometer deflection steadily at zero, and the tem
perature of the bath was again read on the same mercury thermo
meter.
.11' the aperture is too large, oscillations of the gas take place and
•ilvanometer deflection is not quite steady, the initial deflection
being somewh tcr than the true value. If the aperture is .too
narrow, prolonged expansion resull be process is not adiabatic.
Tn pr; rture was gradually diminished and when over
shooting was eliminated the deflection was instantaneous and per
steady. The atmospheric pressure was read on a Fortin's barometer
and y calculated from the foregoing formula, y was found to be
1.4034* at 17°C. This method cannot be used at high temperatures
since it is impossible to determine accurately the cooling correction.
35. Ruchardt's Experiment. — A simple method for determining
y, which is suitable for classroom demonstration, has been described
by E. Riidiarrit, The apparatus consists of a Large glass bottle V
(Fig. 16) fitted airtight with a glasstube at the top and a stopcock
H at the bottom. The glasstube lias a very uniform bore in which
a steel ball of mass m fits very accurately If the ball is dropped into
the lube, it begins to oscillate up and down and comes to rest after a
few oscillations. If the period of oscillation be determined with a
stopwatch, y can be easily calculated
Let A be the crosssection of the glass tube, v die volume of the
bottle, b the barometric pressure, and p the pressure in the flask.
Then in the equilibrium position
,I + 3
* Using this, the velocity of sound in dry air at 6"C was calculated fn a
equation (3fi) to be 331.38 metres which is in close agreement with Hebb's mean
value 331.41 m./sec. (Sec 37).
1
VELOCITY OF SOUND METHOD
57
ball now moves a distance x downwards it compresses the
diabatically increasing the pressure to p + dp, hence ,tt equatum
. diabatically
ol motion is
m  — Force of restitution
= Ad;>.
pv* — constant, we have
dv Ax
■ypyvPT
(35)
i =Ax.
d*x
s
•ii .• lich the period
of oscillation comes
T = 2
y =
v,
4ir •■• i
(W)
Fig. 16— Rfidiardfs
Apparatus.
pA*T* m
I litis, knowing T_, p and' the constants of the apparatus, y can be
aked.
(2) Velocity of Sound Method
36 This method also depends upon the adiabatic expansion i
• ,.'. , a gas but differs from the foregoing method in that
direct measurement of changes in temperature or pressure need
io be observed. The method has given us the most accurate data
regarding specific heats Cor both high and low temperatures and so
all consider it in some detail.
The velocity of sound in any Kind is given by the equation
U — ^nrfe where E s p is the adiabatic elasticity of the fluid and
density. For adiabatic changes in perfect gasesf pv v . constant
(p. 48) , hence
Ef =  B fi} tf tan (35).
U
V
hp
(38)
Thus, if we determine the velocity of sound in the gas we car*
37. We may adopt either of the jpo following methods. The
;ite velocity of sound in the gas may be determined, or we may
*See Barton, Sowd, , . . , ,_,
ir real gases we must take into account the true equation of state [Lhap.
c the value of \/.:pi'iiv), from that equation. In all accurate work
d in.
&s
CALOHIMJE.TRY
compare the velocity with that in another gas (say, air) which has
been determined accurately by other methods.
For our purpose we discard the largescale determinations of the
city of sound in air on account of the various defects inherent
in them. The most, accurate direct determination of the velocity of
; 'itcl in air was made by Hebb in 1905 by a method depending on
reflection of sound of known frequency from parabolic mirrors,
His mean value after employing ail corrections gives 33141 r
sec, as the. velocity of sound in air at 0°G and 760 mm. pressure.
Now we must remember that practically all determinations of the
velocity of sound in gases have been made in tubes, but the velocity
in ;:. tube is not the same as in free space. Corrections have to be
applied to reduce this velocity to that in open space as explained in
the next section. In equation (37) the velocity in open space must
substituted. Dixon has directly determined the velocity of sound
in different gases from 15°C to 1000°C in a very satisfactory manner.
5 result may be employed to give y.
Method based on the Measurement of Wavelength.
3S. Kundt's Tube. — Kundt first devised an apparatus by means
of which he could find the velocity of sound in a gas. This consists
simply of a glass tube about 1 metre in length and diari eter.
One end of the tube was fitted with a mo opper, while through
b SrE s dl^l U^
17,—. Kundt's Doubletube A]
looselyfitting disc: carried by a glass or metal
: ingrod 11 clamped at its centre. Later Xundr
employed the doubletube apparatus indicated in Fig. 17. Two tubes
art; connected by means of the soundingrod S which as
; l] . 1 at distances onequarter of its length from either end. The
.bber corks., d, d in the tubes A and B provide the damping
ngement. The pistons P. P can be moved to and fro lo bring
die tubes in resonance with the rod S. Throughout, the length oi
each tube is spread some light dust such as Ivcb podium powder or
silica dust. One tube is filled with air and the other with the experi
mental gas. The soundingrod S is excited by rubbing it at the
re when the dust is thrown into violent agitation" at the anti
nodes and collects at the nodes. The distance between successive
nodes equals ha If wavelength, and knowing the frequency of the sound
the velocity is easily obtained. The double form of the apparatus
*l very convenient for comparing die velocity of sound in anv ihts
,'ith that in air for '
w
. (39)
VELOCITY OF SOUND METHOD
59
precautions are, however, necessary. The tube and the
, us be perfectly dry. Carefully purified air must be used
n for the various impurities must be made. Too much
, not be used for excess of dust diminishes the ve loci y.
in the diameter of the tube diminishes the velocity.
m the velocity in the tube the velocity m open space must
reed. Though mathematical equations giving the req m
. on have been developed by Helmholtz, Kircbholt and others
re not quite adequate and the best method is to express the
U' in the tube as
U' = U(lkC), . . . ♦ (40)
where V : velocity in open space,
h = a constant depending on the tube (its radius, thickness,
the thermal conductivity, surface, frequency of the sound,
etc.)
C — a factor depending on the gas (its viscosity, density, ratio
of specific heats, etc.)
Kirchhoff showed that
C =
/fl'Ww)!
here » = viscosity, p = density, Y r= ratio of specific heats and
bte K being the thermal conductivity. For air from Hel ■■■
riments we know U } and by observing U[ and calculating C, k
h tube is determined. This value of k is employed to give
the velday in open space for any other gas.
Kundt and Warburg later employed this method to determine
the velocity of sound in mercury vapour. One of the tubes contained
ury vapour and was heated in an airbath to about 800 C, The
distance between two nodes was measured when the tube cooled.
They found y— 1.666, Ramsay employed this method to find y for
argon but on account of certain difficulties he got a low value. Behn
and Geiger improved the apparatus considerably. They dispensed
with the sounding rod and employed a sealed tube containing the
experimental gas as the source of sound. This tube was clamped in
its middle and was excited like the soundingrod. The tube should
be chosen properly and its length be adjustable so that the contained
gas may give resonance with "the sound emitted by die rod. T he
apparatus" was eminently suited for gases at high temperatures. This
method was later>mployed by different investigators particularly by
Partington and Shilling.
* It can be easily shown from very simple considerations that for a mi*
of perfect cases,
_a_ = h + ii_
r— 1 yi— 1 72—1
•where f h £s are the partial pressures^ of these gases, P the total pressure and Y
stands for the ratio C P /C r for the mixture.
60
CALOR1METRY
[chap,
39. Experiments of Partington and Shilling. — These investigators
determined the velocity of sound in various gases up to 10G0 D ~G bv
a resonance method. The apparatus is diagram m a tically represented
in Fig. 18.
I'F is a silica tube 230 cm. long and wound over almost
along its entire length with heating coils. To this tube is attached
it X a glass tube, MM, 150 cm. long. Inside the former is the piston
1 J of silica carried by the rod A, also of silica. BB is a steel tube
joined to A by means of a cork. The tube BB carries a saddl
18.— Partingtan and Shilling's ,'■•
ing on a millimetre scale, thus die displacement of the piston
Through this tube pass die thcimo couple leads to the
potentiometer system £. The other end of the silica tube is closed by
a telephone diaphragm T which can be moved by means of the screw
Y. This end i losed gastight by means of the belljar J. . The
diaphragm is excited by a valve oscillator V giving a note
of Erequency 3000. D is a sidetube from which a rubber tube leads
to the ear of the experimenter. X is ati asbestos phie to prevent
radiation of heat to M. v
The silica tube is filled with the experimental gas and maintained
at Che desired temperature. The central tube AB is gradually moved
• from 1} and the successive positions of the saddle I on the
millimetre scale corresponding to a maximum sound in D are noted.
The successive distances correspond to A/2 and knowing the frequency
the velocity is determined. The position of T has Co be adjusted at
different temperatures in order to give maximum sound in D when it
will be at a distance A/2 from the "latter.
nno ?°°k employed this method to find y for air and oxygen from
SO to p3 £. His apparatus may be visualized if we imagine the
hotair bath of Kundt and Warburg to be r.huvd Uv :i h; ...
flask containing liquid air.
specific; heat of superheated vapour
01
■ i 1 1 1 i
40. Specific heat of superheated or nonsaturated vapour.— Reg
_.£ determined the specific heat of superheated or nonsaturated
id other vapours with an apparatus which wa itially
liar to that shown in Fig. 12 (p. 49). Steam is superheated to
by passing it through the spiral S in the oilbath kept at a
.Lure above I06°C, The superheated steam is next passed
n constant pressure into the condenser kept immersed in the water
trimeter, and the rise in the temperature of the latter from 8 X to a
noted.
If m h the mass of steam condensed, e p its mean specific heat
. rant pressure between the temperature T and its condensing
tnt 0° at constant pressure, and L the latent heat of steam at d°>
the quantity of heat given by the steam is
Ulcers) +mL+m{&8J,
This must equal a; (0s fli) where w is the thermal capacity of the
lori meter and its contents. Hence
]TB)+mL+m{e6 s )=w{6 s e t ). . . (41)
experiment is then repeated with another value of T and a
►n< ' n similar to the above obtained. Solving the two
qu the wo unknowns c p and L are determined, Regnault
thus found c t for steam between 225*C to 125 L 'C to be 0.48.
In case of vapours of other liquids, the specific heat of the liquid
must be taken into account in writing out the above equations.
41. Results.*— In the foregoing pages we have considered the
various methods of finding specific heats. In tablef 5 (p. S2) we
e the values which best represent Lire experimental data. We
reserve our comments on these values for the next chapter.
replaced by a long Dewar
 Taken from TLmdhuch tier E.vpfn>r<e> k, Vol. 8, p. 33.
62
CAIORIMETRY
[chap.
Table 5. —Molar Heats in Calories at 20 °C and
Atmospheric Pressure,
Gas
Cp
e.
Remarks
Argon
Helium
4.P7
2,98
1,666
>
4.97
2,98
1,666
t Monatomic
tlrogen
6.865
4,88
1,408
1
Oxygen
7.03
5,035
1.396
Nitroj
5.93
4.955
1.402

Nitric oxide
7.10
5,1.0
1.39
Diatomic
Hydrochloric acid , .
7.04
5,00
1.41
Carbon monoxide . .
6.97
4.98
1,40
Chlorine
8,29
6,15
1.35
Au
6.950
4.955
1.402
Carbon dioxide
8,83
6,80
1.299
I
Sulphur dioxide . ,
Hydrogen sulphide
9.65
7,50
1.29
 Triatomjc
8.3
6.2
1.34
.
8,50
6,50
1.9]
\
i' ili
12.355
10,30
1.20
lene
ene
10.45
10.25
8,40
8,20
1,24
1.125
r Polyatomic
8.80
*&!
1.315
42. Special Calorimeters.— Various types of calorimeters have been
devised for special purposes e.g., for the measurement of the heat
of coin bastion, heat of chemical reaction, heat of dilution, etc., but
they involve no new principles. Particular interest, however, attaches
to the determination of heat of combustion in industries, for the
value of fuel u judged mainly from its calorific value. This heat
can be easily determined with the help of the calorimetric 'bomb'.
Fig, 19 indicates Che calorimetric 'bomb*. It consists of a stout
elcylinder A fitted with a cover held down tightly bv suitable
means. The cover has a milledhead screw valve which varies the
cavity A a and thereby regulates the admission of oxygen through the
tubes B and C into the bomb. Through the centre of the cover but
insulated from it passes the wire i which is connected to die platinum
wire wj the other end of the latter being connected to e. There is
another similar screw valve varying the cavity k, through which as
«] go out of the bomb. To enable the bomb to withstand the
osive action of the products of combustion it is plated inside with
gold, though platinum would be better. The bomb is enclosed in an
HEAT BALANCE IN THE HUMAN BODY
calorimeter such as is used for the method of mix!
ilorimeter is provided with a
and accurate mercury thermo
. i . I he whole is surrounded by
n.l temperature jacket.
To find the heat capacity of the
I. Ir and its accessories a known
mnt of electrical energy may be
iii in the system or a fuel ol
calorific' value burnt. The
method is adopted in
ing laboratories and the
t in actual practice. Benzoic
id is most suitable* for this cali
bra t ion. The fue 1, if solid, is formed
i ;all briquette; if liquid, it is
ul i d in pure cellulose and put in
I he platinum dish F and ignited.
mi three times the amount of
just necessary For complete
combustion Is admitted through B,
l"' oxygen is generally employ
cd ;n a pressure of about 25 atmos
and at this high pressure the
obustion is almost instantaneous,
,i result of these experiments it
en found that the calorific
i »f anthracite coal, wood (pine),
Lrol and methylated spirit are 8.8, Fig. 19— The Calorimetric Bomb.
i.l, 11.3 and 6.4 kilocalories per gram respectively.
43. Heat balance in the human body. — The temperature of the
human body remains almost constant in health. The chief loss ol'
heat is from the skin and from the excretory functions, the skin
niracting in winter to diminish this loss. Heat is supplied to the
ly by the food we eat and by the oxidation of living tissues and
i ides. The blood stream serves to keep the temperature of the
ody uniform. Due to the larger heat loss from the body in winter
have to take more food and cover ourselves with heavier clothes.
Books Recommended
!. Glazebrook, A Dictionary of Applied Physics } Vol. I, article
on 'Calorlmei
Partington and Shilling, Specific Heats of Gases*.
3. Handbttch der E^perhnenlolphysih, Vol. 8, Part T.
CHATTER HI
KINETIC THEORY OF MATTER
The Nature of Heat
1 Historical,— In the legends of some ancient nations, it is
iade for man by some friendly spirit by rubbing
re bis
increase
be highly
any process
the cailv~philosophers had no correct notion aooui iu ^J p£*lo
"ophSd I oin the observation that heat could pas, spontaneously from
a C body to a cold one. Heat was, thereto, supposed to be a
kind oE fluid— the caloric ftutd.
Various fictitious properties were assigned to this JOT*™""
fluid. It was supposed lo possess no weight, since bodies did not
rrLe in weigh/ on mere heating. Further, it was supposed u>
elastic allpervading, indestructible and uncreatabk by
' The particle., of this fluid were supposed to repel one
another strongly which explained the expansion of bodies when heated
d l" o tlielniission of Seat during combustion. Temperature wi
, ;,r to potential or level. When the body was heated the caloric
id was supposed to stand at a higher level than when cold Pro
of heat bv friction was compared to the oozing out of water
™ m a sp , Th e caloric fluid when thus squeezed
out n I itself as i
Doi caloric theory of heat began to be thrown cowards
Leenth century. The earliest philosopher to have
. ,.u:al nature o£ heat was Count RumfonL*
In those da a:rc made by casting solid cylindrical preces and
inside by a boring machine, Rumford in 1798
hat apparently an inexhaustible amount ot heat could be
prod the friction of the spindle o£ the boring machine against
body of 'the gun, though the amount of iron scraped was very small
lie undertook protracted experiments and found that the amount
of heat produced (measured by the raising of water to a high tem
perature) bore no relation to the amount of iron scraped, but was
proportional to die amount, of motion lost. He henceforth rejected
the caloric theory and asserted that Heat is only a kmd of Motion,
Whenever Motion disappears it reappears as Heat and there js an
£xact proportionality between the two. He even made an estimate
of what we now tali the Mechanical Equivalent of Heat. His value
: not much different from the value now adopted as standard.
* Count Rumiord (17531814) was born in North America Rein r loyal to
Grr , ..,  i:r [ :T: , the American War of Independence, he had to flee from
bis country. He entered into the service of the Prince of Bavaria aiui
I in charge of the arsenal at Munich when he perforated the ;.■:•!, vatec
••, die bonus of guns. Tn 1799 he went to London and was one
of die Sounders of the Royal Institution,
Face \h 64,
James Pkescott Joule (1818— 1880)
(P fc)
Born near Manchester, Joule was educated at home. The main work
which occupied the greater part of his life was on the relationship
between work and heat. He established the principle
of the mechanical equivalent of heat
James Cleek Maxwell (1831—
Born in Edinburgh, educated at Edinburgh and Cambridge. He became
Km* a Colkge, London, in i860 and Professor of Experi
mental P hy5fcs ■ charffe of ^^ ,£££*
Abridge, «n 1871. Hi, greatest work was in con
necbon with the development of the kinetic
theory of gases and the foundation of the
electromagnetic theory of light.
(P. 75)
FIRST LAW Of THKRMODVNAM1CS
65
In 1799. Davy showed that when two pieces of ice were rubbed
gether water is produced. It was admitted by all that water lias
greater quantity of heat than ice. I Now supporters of the caloric
; i v asserted that heat is generated in friction because the substance
produced by friction has less capacity for heat than the original subs
tance. But the substance produced in Davy's experiment (water)
has greater heat capacity than ice, hence the caloric theory became
untenable. Davy's experiment proved the greatest stumbling block
for the caloric theory.
But the valuable work o£ Rumford and Davy was soon forgotten
nd it was *■ njy about forty years later I hat the first law of Thermo
dynamics gained general publicity through the researches of Joule in
'•land, M and ; Lmholtz in Germany, and Colding in
Denmark.
2. Joule's Experiments, — In 1840 J, Joule of Manchester began
his classical experiments for determining the relation between the work
ne and herd generated. We do not wish to describe these experi
!i detail as they, are at present only of historical interest
rhe heal was produced by churning water contained in a cylinder
bv means ol brass paddles. This could be kept revolving by means
• l a chm I ilc thread wound over a solid cylinder and passing over
pulJe ' rying weights at either end. The amount of work
i lated by observing the height Lhrough which the weights
fell, i ii' rise in temperature was measured by, a mercury thermo
meter and hence the heat generated could be found. After app]
'.hi ections Joule found that 772 ft.lbs. of work at Manchester
the temperature of 1 pound of water I°F. In 1878 Joule
used ;i modified form of the apparatus in which the work done was
■ the application of an external couple as in Rowland'
to be described later (§5.)
3. First Law of Thermodynamics, — The conversion of work into
i , thus established by Lhe experiments of Rumford and Joule.
e ] the times of Rumford and Joule, the steamengine
n widely applied for various industrial purposes. As we shall see
, this is simply a contrivance for the conversion of heat into work.
Thus it is established that heat and work are mutually convertible.
. ■ may go even further and say that when some work is spent
ng heat, a definite relation exists between the work spent
1 n era ted. These two facts, viz., the possibility of converting
to heat and vice versa and the existence of a definite relation
"ii the two are expressed by the First Law of Thermodynamics.
Matin itii illy, the law may be seated thus: —
If IV is the work done in generating an amount of heat H,
we tii
W = JH, I:
t This was proved by Black'.? discovery of Latent Heat of Fusion. He
equal masses of ice and icecold water alternately inside a room and
observed that the water was raised 4"C in halfanftour, while the ice uiok about
ten hours to melt into water, the temperature remaining constant.
5
—7
66
KINETIC THEORY OF MATTKU.
:
where / is a constant, provided all the work done is spent in p; i
hie heat and no portion is wasted by fricdon, radiation, etc. If H
U expressed in calories and W in ergs, J= 4.186 X W The truth
of the second statement embodied in the law is amply proved by the
fact that Lhe various methods for finding / (sees. 4—7) yield almost
identical values,
4. Methods for determining ./.—Various methods have been d<
ed for finding the value of the mechanical q > of heat hut
the method of fluid friction and the electrical method are the only
ones capable of yielding accurate results, and hence only these will
be considered in 'detail. There is, however, an ingenious method of
calculating the value of / which was first given by J. R. Mayer in
1843. From the theoretical relation J(G P  C v ) — R (p. 46), he cal
culated the value of /. Thus, Eor hydrogen R = p*VJT» — 8.3 14 X W 1
ergs per mol per °C (pp. 6, 10) and C p C,= 1.985 cal. per mol per
°C (p. 62) .
8*314xlQ 7
.*. J— — 1985 ~ = 4  T8 X 1{)7 er S s P er calorie.
Certain other methods* that have been employed are enumerated
:
(I) Measurement of heat produced by compressing a gas— Joule.
Heat produced by percussion — Him,
(3) Work done by a steamengii
This wi Him in 1862. He measured the amount
the cylinder of the steamengine in a
at a known temperature and pressure. The
total heal y the engine was found by conducting
ill >• j i>: steam into a calorimeter, and the heat loss due
to ci r causes was estimated. Thus the net
"••■Lint of heal; which is converted into work is obtai I
ik done by the engine was found from an indicator
din;; ram (CI ap. IX.). Equating these two Hirn got a
value of .7 = 4.18 X 1° 7 ergs P er calorie.
(4) Heat developed in a cylinder kept stationary in a rotating
magnetic field produced by means of p ilyph: ie alternating
I ictric current — Bailie and Ferry,
5. Rowland** Experiments.— Joule's thermometers were not :
dardised and thus errors of 1 or 2% may arise from this cause. Tiie
rate of rise of temperature in his experiment was rather slow (about
1 per hour) and hence the radiation correction was :
Rowland minimised this source of uncertainty by designing a special
apparatus with the object of securing a rapid rate of rise of tempera
ture (40°C per hour) , the principle of the method being identical
with that of Joule.
♦ For a ' '.'.l a$ Llit: methods see Glazebrook, A Dictionary of Api
,. V ■!. I, p. 480.
in
ROWI AND S EXPERIM ENTS
67
'I fur calorimeter was firmly attached to a vertical shaft ab [Fig.
ch is fixed a wheel kl wound round with a string carrying
;it either end, the whole being suspended by a torsion

A.
Fig. 1 (a). — Rowland's Apparatus.
wire. The axis of the paddle [Fig. 1 (b) ] passed through the bottom
of the calorimeter and was attached to the shaft ef. The latter could
>tated uniformly by the wheel g driven by a steamengine. The
number of revolutions jjyas automatically recorded on a chronograph
worked by a screw on the shaft ef. The revolution of the paddle at
an enormous rate tended to rotate the calorimeter in the same direc
tion on account of fluid friction. This was prevented by the external
couple produced by thi ■ <;. p and the torsion wire. For the
' urpose of accurately determining the radiation correction a water
acket tii surrounded the calorimeter.
The paddle is indicated separately in Fig. 1 (c). To a hollow
cylindrical axis four rings were attached, each having eight vanes.
68
KINETIC THEORY Or MATTER
CHAP.
Around these were the fixed vanes, consisting of five rows of ten
each, which were fixed to the calorimeter. Thus the liquid could
T£<
Fig. 1 (b).
F3g.l(tf).
b* vigorously churned. The rise of temperature was recorded by a
thermometer suspended within the central sievelike cylinder in which
water circulated briskly.
If D denotes the diameter of the torsion wheel and mg, mg, the
ded, the work W done in » revolutions of the paddle
is given by
W — couple X angle of fcwist — ffigD 2/ttfi. . , {*•)
the thermal capacity of the calorimeter and its contents,
perature i Ldiation) , the heat produced
by fri MB8, hence
(3)
2mt. tngD
3 — MBd~
If D is in an, and mg in dynes, / conies out in ergs per calorie.
Corrections were applied for the torsional couple, for the weights
in air which must be reduced to vacuum, for expansion of the
til wheel, etc. Rowland found J =4,179 X lu? ergs for the 20°
calorie. La by recalculated from Rowland's observations by applying
corrections and obtained the value / = 4.187 X 1° T er g s * or ^ 15 °
calorie.
Reynolds and Moorby obtained bv a modified apparatus, the value
of the mean calorie between 0° and 100 *C to be 4.1833 X 1 T ergs.
Hercus and Laby employed what is in principle an induction motor,
to find J, and obtained the value / = 4.186 X *0 7 ergs per caloric.
6. A Simple Laboratory Method of finding J.— For laboratory pur
a simple apparatus for finding/ due to G. F. C. Searle is
in Fig. 2." but the accuracy, attained by this apparatus
is not great. A is a brass cone held "rigidly in position by means of
in.
ELECT] "R I £ : AL METHODS
69
Fig. 2, — A simple laboratory apparatus
for /,
nonconducting ebonite pieces attached to a brass cylinder C, which
am be made to revolve by means of a motor. Inside A is another
; cone B fitting smoothly into it and attached rigidly to a wooden
disc D. The latter has a groove
running round its circumfer
ence and carrying a cord which
passes over the pulley P and
supports a weight mg. When
die outer cone rotates rapidly
die inner one tends to move in
the same direction on account
of the friction between the two
cones, but is held in position by
properly suspending a suitable
weight mg at the end of the
cord. Tire inner cone B con
tains some water, a thermo
meter and a stirrer (not shown) .
When the weight mg ^ is
kept stationary the turning
moment exerted by it just, bal
ances the frictional couple. If
D is the diameter of the disc
the frictional couple is mgD/2
and die work done by it in n
revolutions of the cone is 2nir.mgD/2. If M be the water equi
valent of the cones and the contents, $0 the rise of temperature pro
duced by friction, then
nirmgD — JmlQ, . ,  . (4)
whence / can be calculated.
7, Electrical Methods.— These methods have already been des
cribed (pp. 3741) in full detail. It is easily seen that if the specific
heat of the liquid be known, this method gives the mechanical equi
valent of heat. There are tw r o methods: — (1) Steadyflow method,
(2) Rise in temperature method. The former was employed by
Callendar and Barnes who did their experiments with great care and
skill but the principal source of uncertainty in their work lies in the
value of the electrical units employed. According to Laby we may
put the E.M.F. of the Clark cell used by Callendar and Barnes as
1.4335 volts at 1&°C". Tlie*y employed the international ohrn which is
equal to 1.0005 X 10° e.m. units. Reducing Calendar's results with
the help of these data Laby gets for the mechanical equivalent of
20° calorie a value of 4.1795 X 10 7 ergs. This yields 4.1845 X J0J
ergs as the equivalent oE the 15° calorie. The most accurate experi
ments on the subject are those of jaeger and Steinwehr by the rise
of temperature method, and of Laby and Hercus by the mechanical
method, the respective values being 4.1863 X W a nd 4.1860 X 10T for
the 15° calorie, Osborne, Stimson and Ginnings have recently found
70
KINETIC THfcORY Ol? MATTER
CHAP,
i In: value 1.1858 by the electrical method. Hence, we can adopt the
value 4,186 >< 10 r ergs P er calorie.
Exercise. — Joule found that 778 li.lbs. of work can raise the
mperature of 1 pound of water IT'. Calculate the mechanical
equivalent of heat in C.G.8. units.
778 f L. lbs. =; 778 )< 30.48 )< 453.6 gm. X cm.
= 778 X 30.48 X £53.6 X 981 == l«55 X 10" ergs.
Heat produced = 453.6 X $ = 252 calorie.s.
,\ J =z 1.055 X 10i*/252 = 4.187 X 10 T ergs per calorie.
8. Heat as Motion of Molecules. — From these experiments it is
[early established that heat is a kind of motion; the next question is
—motion of what ? The answer was given by Clausius and Kronig in
1857 for the case oL gases. They said thai onsists in the motion
of molecules or the smallest particles of matter.
The idea that if we go on dividing matter we ultimately come to
imall particles which cannot be further subdivided dates from very
ancient times, But it remained a barren speculation till Dalton gave
to it a definite form in the middle of the last century. The history
of the molecular theory is known lo all our read md n< i ount
Et need be given here. that according to it all
1 oE a large tiumbei of molecules, all molecules of
Line substance being exactly idem regards ■•, etc.
ii. En the solid and liquid states these mole
while in gases they are Ear apart from one
her.
to i lie Kinetic Theory of Matter hear, is supposed
to consist in the motion of tin ecules. The identification ol heat
with motion of molecules is not a mere hypothesis. It is able to
explain and predict natural phenomena and at present there is little
i solid foundations of truth.
9. Growth of the Kinetic Theory. — The Kinetic Theory of matter
two fundamental hypotheses : (1) the molecular structure
ol matter, (2) the identification of heat with molecular motion. The
1 1 'St of these was established early in the 19th century while the second
established by the experiments of Rumford, Joule, Mayer and
Colding. We may, however, consider Daniel Bernoulli! (1730) as the
founder of the modern kinetic theory as he was the first to explain
Boyle's law by molecular motions. Clausius and Maxwell in the
middle of the 19th century placed the theory on a lirm mathematical
basis. Among the other prominent contributors to the theory are
Boltzmann, Meyer, Jeans, van der Waals, Lorentz and Lord Rayleigh.
Up to the beginning ol the present century, however, the theory
had been developed entirely from a mathematical standpoint. There
no direct experimental proof of the actual existence of these mole
les or of their motions. Gradually, however, much evidence has
•Since the discovery of isotopes tlii: remark requires some modification.
in.
EVIDENCE OF MOTJttL'LAR AGITATION
71
accumulated in favour of these views, the most important being the
Brownian Movement phenomena investigated by Pcrrin in IW&.
10. Evidence of Molecular Agitation. (1) The phenomena of
ssrtsssssse: &
nde o C0 2 . Alcohol
„.,. d i ffus es into the entire mass or : ,
water E«n sold is found to diffuse into leaO.
k) A jeu tends to expand. It is a common
experience that the bh ufcs of the gas
'd To dv away and FT^jZ
ansibility. T& rectilinear motion ol the mole
culef is 'ruled by the experiment of Dunoyer in
w hi, obtained atomic or molecular y;v
t™ annus consists of the tube ABC (Tig.
I" u Compartments, A, B, C by p ***
iimhri«ns 1> and F. The apparatus is highly e\a
SS2 *fr ic .idcBibe J. The end F contain
£ heated to abou Ift^S
* through circular apertures in 1 and
deposited on . tamd J f ] ;
the second hole, form . ' h ^
holes bv straight lines, which proves that the mole
1 : straight lines.
. Phenomena of evaporation and vapour
14')' The m s I aw s ca n 1 >c dedu red from tli e kinetic
theory'". . 13). Other results ob from the
, as suecihe heat, iher
molecular diameter, etc., agree with experimental results or deductions
from other branches of Physics.
(5) Phenomena of Brownian movement.
11. Brownian Movement.* This phenomenon was first discover
cd bv the English botanist brown in 1827 while _ ol
aqueous suspensions of fine inanimate spo u der high powei
,pc. He found the spires dancrng about in the wild
I, ion. The phenomenon can be readily observed d small particles
ided in ; as in a colloidal solution, are examined n
, ■ ■ ... __ u A *. n » .. n Jn» 'in iiUr^.mirrosrnTkf. the
A
F
l\ . " 
Dtmoyer's
apparatus.
■■■ f, [ detail di ' siom see the Authors" A Treatise or. Heet, 3.4S— 3.4S,
72
KINETIC THKORY OF MATTER
CHAP.
V/
I Vilhittjih^J"
The Browniari movement never ceases it is eternal and spon
taneous, and is independent of the chemical nature of the suspended
panicles, all particles of the same size being equally agitated. Smaller
particles are, however, much more vigorously agitated than bigger
ones. The motion becomes more vigorous when the temperature is
aaed or a less viscous liquid is chosen. It is just perceptible in
glycerine and very active in gases. No two particles are found to
execute the same motion, hence the motion cannot be due to any
convection or eddy currents.
Th,e discovery of such spontaneous motion, and the fact that the
motion is maintained even in viscous liquids without die application
Oi my force was a great puzzle to earlier observers. Gradually, how
ever, it has been established that the Brow ni an movement is due
to the impact of the surrounding molecules of die liquid on the
Brown jan panicle. It is evident that the
Aficroscoj* forces due to molecular impact, will almost
completely balance if the size of the particle
is very large (say, a large body immersed
in the liquid) but there can be no balance
if the size is small Any small particle will,
therefore, be acted on by a resultant un
balanced force .and will consequently exe
cute motion. As this force varies at
random, so the motion of the particle "'.'■
be at random and will be some*'
Fi K 6 (p. 81). Thus the phenomenon
aent is a d of the existence of w i
ion.
The study of the kinetic theory is best approached through a
ases. The kinetic theory of the liquid and solid states is
comparatively undeveloped and will not be discussed m this book.
12. Pressure Exerted by a Perfect Gas.— It has been shown above
that a gas consists of molecules in motion. As a consequence, it
wre on the walls of its enclosure. To calculate this
pressure we first make several simplifying assumptions. These are
the following ;. — .
(1) Though the molecules are incessantly colliding against one
mother and having their velocities altered in direction and magnitude
each collision, yet in the steady state the collisions do not affect
the molecular density of the gas. The molecules do not collect at
one place in larger numbers than at another. Further, in every
1 nient of volume of the gas the molecules are moving in all direc
tions with all possible velocities. The gas is then said to be in a
state of molecular chaos,
(2) Between two collisions a molecule moves in a straight line
with uniform velocity. This is because the molecules are material
bodies and must obey the laws of motion.
Emufok
viug
ill.
PRiiSSL'RK EXERTED 1SY PERFECT GAS
7$
(3) The dimensions* of a molecule may be neglected in com
parison with the distance traversed by it between two collisions, called
its free path. The perfect gas theory treats the molecules as mere
masspoints.
1 1 The time during which a collision lasts is negligible com
pared' with the time required to traverse the free path.
(5) JThe molecules arc perfectly elasticf spheres. Further, no
appreciable force of attraction or repulsion is exerted by them on
one another or on the walls, i.e, ? all energy is kinetic. This is proved
by Joule's experiment (p. 47).
We now proceed to calculate the pressure exerted by such a
gas. We will employ the method of collisions, because it is very
simple.
Imagine a perfect gas enclosed in a cube of unit sides and con
sider a molecule moving with the resultant velocity c and component
velocities «, v, w along OX, OY, OZ axes respectively. " The axes
are taken to be parallel to die sides of the cube. The molecule
collides with the surface of the cube perpendicular to OX with the
velocity u. From the principles of conservation of energy and
momentum it follows that it will rebound with the same velocity.
Hence the change in momentum suffered by the molecule during
collision is %mu. The molecule strikes that particular surface «/2
times per second, hence the change of momentum per second is
2imi.u/2 = ?nu s . Since pressure is equal to change in momentum
per second, the total pressure exerted on that surface is "Zmu 2 where
the summation extends over all the molecules.
/. p x — Xmu*.
• (5>
Now "'— = w" J where u is the average of u 2 over all the n
molecules.
,\ p K — mnu 2 ..... (6)
Similarly, the pressures on the other surfaces are
p y = mntJ 2 , p z = mnwK
*If we consider the dimensions of the molecule and the forces of attraction,
we gel van der Waals' equation (Chapter IV).
t The assumption of perfectly elastic collisions, on the average, is warranted
by the fact that we can convert into work all the heat supplied tr> a perfect
gas. For otherwise addition of heat would increase molecular velocities ami
hence also the force of collision, and if deformation of molecule results, all beat
may not be converted back to work. The picture here given is essentially that
of a monatomjc molecule; there will occur deformations of polyatomic molecules,
accompanied by an increase or decrease of rotational and vibrational ci
but on the average there is no net los3 or gain of translational energy during
collision. Equation (9), however, can be deduced without the assumption of
perfectly elastic collision. For details see the Authors' Treatise on Heat, Sec.
3.12, footnote 2.
74
KINETIC THEORY OF MATTER
': CHAP,
where a» a , ar 2 denote the mean square velocities in the other two
perpendicular directions. Since experiments* show that p x= p^=^&,
we have
u % z=^iP —~w2. n\
This is also to be expected from the fact that the molecules do not
tend to accumulate in any part of the vessel. But
n n
where c 2 is the resultant mean square velocity. Hence from (7)
and (8)
IP = £^
and finally (6) yields
p = $mnc*. (9)
But mn = Ps the density o£ the gas, since ra is the number
per ex.
p = y <>\
(10)
(11)
where £ is the kinetic energy per unit volume. Thus we see that
the pr. a perfect gas is numerically equal to twothirds of
kinetic energy of translation per unit volume.
ie molecule as if it suffered no collision
i others. If a molecule encounters another it gives to
ther which bv ;, h as t0 traverfle
tance as the first one if there had been no
tion (■!) no time is lost in this collision and
ur calculation holds true for a perfect <ras.
(10) enables us to calculate the' mean squ
Kity c" of the molecules of any gas, for 1?=. $pf p . The pressure
;1 " ; of a gas can be found experimentally and hence c=
calculated. Tims the density of nitrogen at 0°C and atmosp!
pressure is 000125 gm. per cc Hence for nitrogen the root me
square velocity!
C =
V* \/
3 X 76x1359x981 iM
rjm  = 4.98X10* an./sec.
*Thk is so for very email cubes. Rigorously speaking, for large cubes
tht SSL " "* to k ***** m aC ^ of sSvitation^T ptS on
is^mult may b c easily deduced from the laws of elastic impact
? C is the square root of the mean square velocity and differs from the
m and 20 i s Ta whale the root mean square of 10 and 20 is
/HP +20* _ .
V 2*~" =1S ' 81 ' Thi: exact relation between C and c for the molecules
of a gas in equilibrium h Riven later (§16),
ill.
DEDUCTION OF GAS LAWS
75
The formula also shows that the molecules of the lightest gas,
,, hydrogen, would move faster than the molecules of any, other
gas under the same conditions.
13. Deduction of Gas Laws— From the above results we proceed
to deduce the laws of perfect gases.
(1) Avogadro's* Law, — If there are two gases at the same
lire p vflft have from (9)
p = im i n 1 C t a — sJm 2 n 2 C/., . . . . (12)
where the subscripts 1 and 2 refer to the first and the second gas
respectively. Further, if the two gases are also at the same tempera
ture we know there will be no transfer of heat (or energy, since the
two are equivalent by the First Law of Thermodynamics) from one
to the other when ihey are mixed up. On mixing, the two types of
molecules will collide against one another and there will be a mutual
sharing of energy. Maxwellt showed purely from dynamical con
siderations that the condition tor no resultant transfer of energy from
one type of molecules to the other is that the mean translation a]
energy of molecules of the one type is equal to that of the other.
Hence if the two gases are at the same temperature it follows that
}*&* = &#* (13)
Combining (12) and (T3) we get
n^=n 2 , (14)
I.e., two gases at the same temperature and pressure contain the
same number of molecules per c.c. This is Avoga tiro's Law.
(2) Boyle's Law. — Equation (10) states that the pressure of a
,;i. is directly proportional to its density or inversely proportional
to its volume. This is Boyle's Law. This holds provided C a remains
constant which, as shown above, implies that the temperature remains
constant.
(3) Dq Hon' s Law. — Tf a number of gases of densities p ti p.p p%
... and having mean square velocities Ci 2 , tV 2 , C 3 2 ....,. be mixed' in
the same volume, the resultant pressure p } considering each set of
molecules is given by
p  taG 1 % +bA*+WV+' ■
— Pl+P2 + P*+— .♦►♦». (15)
♦Count Amedeo, Avogadro di Quarcgua (1775 — 1856) was born in Turin
where he was Professor of Physics from 1833 to 1850. His chief work k
Avogadro's law,
t Maxwell coiiHidcred the collision of gaseous particles of two different
■ and possessing different amounts of energy. By applying the dynamical
laws of impact viz. conservation of momentum and energy, he found that after
each collision the difference in energy oi the two molecules diminishes by a
certain _ fraction, i.e. the molecule possessing greater energy loses it while that
possessing less energy gains, This process is repeated at each collision, and ulti
mately the energies of the two become equal. For details, sec the Authors'
Treatise on Heat (1958), pp. 845^847.
76
KINETIC THEORY OF MAI EER
CHAP.
ITT.]
MAXWELL S LAW OF DISTRIBUTTON OF VELOCITIES
77
i.e tf the pressure exerted by the mixture is equal to the sum of the
ires exerted separately by its several components. This ia
Dalton's law of partial pressures.
14. Introduction of Temperature. — If we consider a grain
molecule ol the gas which occupies a volume V, equation (10) fields
pF=z$MC 2 , (16)
M being the molecular weight. In order to introduce temperature
in the foregoing kinetic considerations of a perfect gas we have to
make use of. the experimental law, viz., pV — RT. Hence
RT = jMC 2 , x
.... (17)
fV
M
Thus C a is proportional to the absolute temperature which may
thus be considered proportional to the mean kinetic energy of transla
tion of the molecules. This is the kinetic interpretation of tempera
ture. Hence, according to the kinetic theory, the absolute zero of
temperature is the temperature at which the molecules are devoid
of all motion. This deduction is, however, not quite justified since
the perfect gas state does not hold down to the absolute zero. The
interpretation given by thermodynamics is somewhat different
(Chap. X) and is more reasonable. That does not necessarily require
that all motion should cease at the absolute zero.
i put Mz—Nm where m is the mass of a single molecule
the iiuniljer oL molecules in a gram molecule, which is usually
called th>
□ put R/N = k where k is a constant, k is known as
ttoltzrnann's constant. Hence we get
p= (N/V)hT = nkT f
win: notes the number of molecules per c.c. Further from (17)
fcJViiw"  $MT t
or W^PT", . (17a>
Le. t the mean kinetic energy of translation of one molecule is §&T*
Exercise. — Calculate the molecular kinetic energy of 1 gram of
heli: : : N.T.P. What will be the energy at I00°C?
From (17) the kinetic energy is equal' to
Energy at 100°C = 8.5 X 10° X 373/273 ™ 1.16 )< 10*° ergs,
15. Distribution of Velocities. Maxwell's Law. — Tn the above we
were concerned only with the mean square velocity and did not
care to find the velocity of every molecule. But for studying the
properties of the gas further we must know the dynamical state of
the whole system. It Is easy to see that all the molecules cannot
have the same speed for even if at any instant all the molecules

possess the same speed, collisions at the next moment will augment
the velocity of some and diminish that of the others. As the number
of molecules is very large (2.7 X !0 ia per c.c. at N.T.P.) and they are
too small to be visible even in the ultra microscope, we do not interest
ourselves in the behaviour of individual molecules. We treat the
Lilem statistically and apply the theory of probability. We shall
illustrate this by means of an example,' In a big city there are
persons of alleges and we find the number of persons whose ages
lie between definite ranges, say between 10 and 15, 15 and 20, and
so on. So in an assemblage of molecules where the molecules have
all velocities lying between and infinity we find the number of
molecules dn t possessing velocities lying between c and c f dc. In
the steady state this number remains constant and is not modified by
collisions. This number is given by die distribution law of Maxwell
which states :
dn c = 4iraaV**Vi: > .... (18)
ere n h the number of molecules per cc, and a = \/bfir =
mf%vrkT. But we cannot say what the velocity of any individual
molecule selected at random is. We can only say that the probability
that its velocity lies between c and c f dc is '
Thus the distribution law gives a complete knowledge of the gas only
in a statistical sense.
A slight transformation ^putting &c s — x 2 ) will show that the
number dn = 4wjr~&x*e~^d% t which helps us to represent the law
iphically,. Let us plot the function y — £tt~* *""**x 2 against x
ti Curve illustrating;: it
Maxwell's distribution law
Fig. S. — Curve illustrating Maxwell's Law.
(Fig, 5) . Then the number dn of molecules whose speed lies between
7b
KiNirilO THEORY OF MATTER
[CHAP*
x and x 4 dx is proportional to the shaded area. The ordinate y gi
the fraction of the number of molecules posse correspond
ing to x, and £ro.m the curve it is obvious that the probability con
pon< :; — 1 is greatest, while it is considerably less tor % — 2 or
x — \. "Hence we can approximately treaty die whole gas as endov. i
with die most probable velocity corresponding to x — 1.
16. Average Velocities.— We must distinguish between two
velocities, the square root C of the mean square velocity, and the.
mean velocity c. The former is such that its square is the average
of the squares of the velocities of the molecules. Thus
3 UT
■*±l
which we have already obtained in
The mean velogit)
(17).
n
dn,.=
tm
(20)
(21)
The most probable velocity a is that value of c for which N e the
number of molecules with velocity c is maximum. Hence for such
dN
= 0. TMs relation gives «■ Substituting
get
v™=v\
c.
An consequence of the large molecular velocities is
seen in the almost complete absence of an atmosphere from the
surface of the moon. Dynamical investigations show that if a particle
is projected from the Earth with a velocity exceeding \/2gr where
g is the gravity at the surface of the earth' and r its radius, it wfl
never return to the earth and will be lost in space. This critical
velocity Is about II Kilometres per second for Earth and 2,1 kilo
metres for the moon. Russian scientists have recently (2nd
January, 1959) been able to launch a cosmic rocket "Mechta" which
overcame the Earth's gravitational barrier and Hew past the moon
space to become the first artificial planet of the sun with an
orbital period of 447 days,
_ Calculations show that the average velocity of hydrogen at v
ordinary temperatures is about 1.8 kilometres per second and accord
ing to Maxwell's law a large number of molecules have velocities much
greater and also much less than this. Thus all molecules having
than the critical will escape from the planet. Due
certain fraction will always have velocity
to molecular collisions
greater than the critical and will escape. This loss of the planetary
atmosphere will continue indefinitely. It is for this reason that there
LAW OF EQUtPARITXON OF ENERGY VI'
tically no atmosphere on the surface of the moon while the
atmosphere of the Mars is much rarer than that of the earth.
17. Law of Eqnipartition ef Kinetic Energy.— We next proceed
ileal with the law of equi partition of energy. It is hetter to>
introduce here the idea of degrees" of freedom of a system. Suppose
ch an ant constrained to move along a straight line ; it has.
then only one degree of freedom and its energy is given by %mx 3 . If
, allowecrto move in a plane the energy is given by hmx 2 4 hny*.
An ant cannot have more than two degrees of freedom, but a bee
ch is capable of flying has three degrees of freedom, all
translation. Thus a material particle, supposed to be a point, can
have at most three degrees of freedom. A rigid body can, however,
not only move hut also rotate about any axis passing through itself,
The most general kind of rotatory motion can be resolved analytical^
into rotations of the body about any three mutually perpendicular a:
through a point fixed in itself. Hence the degrees of freedom contri
buted by rotational motion are three. We may now state the defini
tion of the term 'degrees of freedom'. The total number of inde
cent quantities which must be known Before the position and
figuration of any dynamical system can be fully known is called
the number :r , :
Now it can be shown from rigorous dynamical considerations that
nergy corresponding to every degree of freedom is the same
as for any other, i.e., the energy « equally distributed between the
various degrees of freedom. This is the law of cquipartition of
kinetic energy and was arrived at by Maxwell* in 1859. Boliamannf
ended it to the energy of rotation and vibration also. It can
further be shown that the energy corresponding to each decree eJ
freedom per molecule is XhT%. This law is very general, but we V
not attempt here to prove it.
Thus if any dynamical system has n decrees of freedom
e^fy I tt at T*K is n )< hTtT, '
18. Molecular and Atomic Energy,— The above theorem is
useful in calculating molecular energy of substances. Let. us calculate
Luc specific heat of uses. In a mm gas the molecules are
identical with atoms and if, as a Erst approximation, we assume the
atom to be structureless point, then from the previous consideration s.
each molecule has got three degrees of freedom and will have tlv»
kinetic energy equal to &X£hT. In the state of perfect gas the
* Maxwell, Collected Works, Vol. 1, p. 378.
fLudwig Boltzmaan (18441906). Born aud educate! in Vienna, he wi
! esaor of Theoretical Physics at Vienna, G raZ Munich and Leipzie On
account ot his ^ fundamental researches he is regarded as one of the founders
of the kinetic theory of gases,
mean kinetic energy of
$ Equation (17a) gives ±mc a = $kT 1 Le. t the
translation per molecule is IkT, If wc assume that the C n<
distributed between the three degrees of freedom, the energy associate* with one
,' : °\ t'.'^om Per mojecule heconins UT. For a formal proof of the
see the Authdhs' Treatise oh Heat, §3.26
Ni:
KINETIC THEORY OF MATTER
CHAP.
molecules possess only kinetic energy and no potential energy. The
total energy E associated with a grammolecule is N times the above
expression. It is thus equal to %NkT =z$RT.
heat at constant volume is therefore
dE
The molar specific
C v =\ T  =^\ R = 2 " 98 &d. /degree.
For all perfect gases we have established the relation
C,~C t
 11.
Therefore, for a monatoudc gas
and the ratio of die two specific heats
J*JL+ Rs!a *±R = 496 cal./dcgrce.
(23)
y =s 5/3 = 1^&
These theoretical conclusions agree with experimental results (see
i able 5, Chap. II) for the nionatomic gases like argon, helium, etc
The specific heat off polyatomic gases can also be obtained by
using the equipartition law. The molecule of a diatomic gas may
be pictured as a system of two atoms (assumed to be points) joined
"dlv to one another like a dumbbell. The sysi
"iti'on to the three components of the velocity of translation of the
common centre of gravity, two components of the velocity of
out two axes perpendicular to the line of centres of the
iras. Thus the system has five degrees of freedom and the
energy E = $RT. Hence
=$R, G,1R, y=14. . , . (24)
This is approximately the case for hydrogen, nitrogen, etc.
(p. 02'). At low temperai • however, C t falls to $R, as Eucken's
•riments with hydrogen show, indicating that the rotation hat.
disappeared. For chlorine C, is greater than ,R. This shows that
Lhe two atoms are not rigidly fixed bur. ran vibrate in a restricted
: nner along the line of centres.
In a triatomk gas a molecule possesses three translational and
•hxee rotational degrees of freedom and hence
C „ = fi >< W = SK, C P =4R, 7 =\M,, . (25)
more complex molecules y approaches unity but is always
greater than it. Tt is not possible to calculate in a simple way the
ergy of internal vibration of such molecules since the vibrations
are not freely and fully developed.
An expression for the specific heat of solid i may also be obtained
from the kinetic theory. We can couse ecules of a solid as
elastic spheres held in position by Lhe attraction of other molecules
and capable of vibrating in a simple harmonic manner about a mean
position. The molecule will have three components of velocity, and
nee three degrees of freedom. The kinetic energy associated with
each degree of freedom is ^kT. On the average die harmonic vibration
i i
MEAN FREE PATH PHENOMENA
81
'•, . ergies and hence the total
issociated with each degree of freedom is kT. T re, the
Eor the three degrees of freedom is $RT f
n ! i heat ia 3i£ = .5.955. This yields Dulong and
 . I .
But the kinetic theory of specific heat is unable to explain the
mi o specific heat with temperature (p. 45), particularly, the
markctr decrease at extremely low temperatures. Further, the
jradual and cannot be explained by the disappearance of
se oT freedom which would involve discontinuous changi
iplcs of ^R. We cannot assume fractional degrees of fvc
the principles of classical dynamics and equipartition law fail
lately. The quantum theory of specific heat has been developed
whu hi existing facts sai tsfactorily.
MEAN FREE PATH PHENOMENA
19. Need for the Assumption thdt Molecules have got a finite
Diameter.' — We have seen in the previous secti ms that the molecules
i i ;i ■'. iving at ordin . iperatures with very large velocities ;
I the case of air it amounts to i ; ; metres per sec. There
is no force to restrain the motion of the molecules. Hence the
was raised that the assumption of such large rectilinear
ies was incompatible with many facts of observation. If the
particles in ing with such enormous velocities., the gaseous n
:ined in a vessel would disappear in no time. But we ar
e that the top of a cloud of smoke holds together for hours., hence
: be .some factor which prevents the free escape of particles.
A very simple explanation was offered by Clan sins. He sftoi id
that the difficulty disap we ascribe to the molecules a Finite
ugh very small volume. Then as :■ i irtirle moves forward, it is
to collide witbl another particle after a short interval, ami its
tty and direction of motion will be completely changed. The
path traversed between two successive collisions will he a straight, line
ibed with a constant velocity, since the molecules exert no fori
ivei one another except during collision. Hence the path of a single
cle will consisi of a series of short zigzag paths as illustrated : 'i
6.
.Some of these paths will be long, others will be short. We
< an define a mean free path A . Add
I the lengths of a large number
paths and divide it by the total
number; this will give ■•'  This
quantity is of peat importance in
tudying a class of phenomena, ca1
port phenomena,, such as vis
cosity, conduction of heat and
ion.
20. Calculation of the Mean Free Path* — We shall give a very
n :• method or calculating d i mi an free path approximately. We
6
i 5— Illustration of h
kinetic: theory of matter
[ CHAP.
the one
m
and
.,!:, the simplifying assumption that all molecules except the one
under consideration arc at rest. The moving molecule will o
with all those molecules whose centres lie within a distance a train
its centre U being the diameter of the molecules), and are thus
contained in a sphere of radius described about tne centre of the
moving molecule. As the molecule traverses the gas with velocity
it wdl collide with all the molecules lying in the region traversed b>
its sphere of influence, The space thus traversed in a second is a
cylinder ot base W 2 and height v, and hence of volume it<**v. .t the
number of molecules per ex. is », this cylinder will enclose ttcl.
itres of molecules and hence the number v of collisions per second
■ n ffff 2 vnj The length of the mean Iree path
V fffl '
1
111 the above we assumed the other molecules to be at rest.
Maxwell corrected this expression by introducing into the foregoing
, itions th( motion of all molecules according to Maxwell
lisi itioi and obtained the result
A— J— W)
■:TTCT~
lumber o isions suffered by a molecule pel
21. Transport Phenomena.— 1 be distribution law expressed b)
o be put in the [otto
dw, . . :
where dn is the number of molecules with velocity romponenl
du, v and v f do, w and zv  dm respectively. 11
nth mass motion represented by
if'' F*+W*)
■^> du dv <m\
dn = n{, L
,,. I icie U = u  U&J v ' '* and W = «  B
11 the gas is not in a steady state any one of the following cases,
:• or jointly, may occur. (1) Firstly, u, x , v 9l xti® may not have
he same value in all parts of the gas 50 that there will be a relative
motion o£ the layers of die gas with respect to one another. We
have then the phenomenon of viscosity. (2) Secondly, T may not
be the same throughout,, then we have the phenomenon of conduc
tion, viz., heat will pass from regions of higher T to regions of lower
T. (3) Thirdly, if n is not the same everywhere we have the case
of diffusion, i.e.', molecules diffuse from regions of higher concentra
te regions of lower concentration. It is thus obvious that
viscosity, conduction and diffusion represent, respectively the transport
of momentum, energy and mass. These are called transport pheno
VISCOSITY
83
ina. These phenomena are brought about by the thermal aeitation
the molecules. But the molecules move 'with very lame velo
cities, wink these processes are comparatively very slow. The
cause of this anomaly lies in the frequent molecular collisions Hence
■ iiiv of these phenomena is most conveniently done through the
chmum of mean free path. The molecules carry with then
tarn associated magnitudes and thereby tend to establish equili
22. Viscosity. We shall first discuss the phenomenon of vis
it)'. Here we shall give an elementary treatment of the pheno
nenon based on considerations of menu free path. Consider a gas
i :>n and choose a horizontal plane xy such that there is a mass
motion , ot the gas parallel to xvplane but no mass motion aTong
K *axis. Assume that the massvelocity „ n increases upwards as z
ases. The molecules above the plane z = z» possess, on the
rage, greater momentum than those below it and hence when
xnles from either side cross the plane there is greater transom*
tentum downwards since the number of molecules movXsl
wy is the same, there being no mass motion parallel to the Vaxis
We can consider every molecule, on the average, to traverse a
equal to the mean free path and then suffer a collision.
le velocity gradient is du /dz the difference in the mean molecular
oss two planes separated by a distance* A is Xdujdz The
1 » oi a molecule being m , the difference in momentum is mUu^fdz.
m. due to heat, motion, the number of molecules moving along the
vax is must be, on the average, the same as that moving akmg the I
the ***«. Hence onetlnrcl of the molecules may be considered
moving along the .axis both up or down, or onlv olS De
he oWrvlr m0 T 1Dg 01 ** "P™*" direction. Consider unit area of
the observation plane z . The number of molecules crossing this area
Si' thf ] " P r rd diimi ? n ** be f ** s ^»ere n /the m,
"Jar density and S the mean molecular velocity corresponding to the
tperature of the gas, since all those molecules contained in^a cylin
der of base. unity and height * cross the unit area in one second
Hence the momentum transferred across the plane in the upward
'ion is lnc[ G mX^) where G is the momentum correspond
to the observation plane. Similarly, for molecules going down
the momentum transferred is ^{c+mX^) . Hence the
il momentum transferred downwards ,, .,;...:. ftp Tllis „
d z
L an accelerating force on the lower layers. Or the lower layers
Vtmily we m us t uke the average reSoIvetl pa rf nf ,
1 '"• ' ■ out to 1 :• instead of A.
Rigorous calculation shows that this will be fytc.
along the  ;
■84
y OF MATTER
•
 retard this fester Layer by a force equal to this. By definition of
;:siu q this force must equal »jd Hence
... (30)
e p is the density of the gas.
23. Discussion of the Result— We have established thai
but X ==— —  (p.
V2 mro*
82) j hence
wo*
Now is proportional to the square mot of the ab
perature, hence the coefficient of viscosity is independci pres
sure or the density of the gas, provided the temperature is co
This deduction is 'rather surprising and was first regarded with suspi
cion. Mayer and Maxwell subsequently showed experimentally that
lw actually held for prcssm: s lyi *een 760 mm, and 10 mm.,
unci this was a striking success for the Kinetic Theory. Bin
both limits the law fails. At higher pressures deviation from the law
ected since the mtermolccular forces ran not be i, At
•e path A gradually increases till ii becomes
arable with the dimensions of th ' then
remains constant Any further decrease of pressure n and
pressures the coefficient of \ iscosity
ire.
Again i) for different gases should vary
is found rue.
24. Conduction.— Let us now find an expression for the
i. In this case th< rature
ies from Iayi i Lo layer and it h the civ
which is tran i one lai mother. Considering the
dUn
the
JnfA
energy gradient " instead of the momentum gradient m •
I transfer of energy downward? per unit area becomes
M_
dz
where E denotes the mean molecular energy pertaining to any
If K be the conductivity of the gas, j— the temperature
client, the flow ol energ) across unit area in the downward direction
is JK ,  where / is the mechanical equivalent oT heat. The
dT f fE dT
j K _ = ^ Xjt _ _
d'E 1
' ' an —
III.]
\ ,\U I O] '..NTS
when c is expressed in heat units. Further considerations show
i above result must be modified into K zeifC v where % is somi
and the variation of c is small, the variation
ol conductn Lth pressure and temperature follows in general the
tine coi variation of viscosity. Thus conductivity like
dependent of pressure. This was verified experimentally
in and others. We shall no* consider the phenomenon ot
on as ii is somewhat more complicated.
25. Valae of Constants. — The value of the root mean square
a gas was calculated numerically for nitrogen on p. 75.
lean velocitj c can be calculated from (21) and is equal to
C 4.93 X I« 4 = 4,5 X 10* cm. /sec. Then (30) gives, on substitm
166 X 10 6 gm. cm* 1 sec',
k ?*i« I0 " fi 9xl0«.
P c ~ 123 :I0" Mv/IlM ,u cm 
i collision* suffered by a molecule per second is
Assuming h = 2,7 x 10" per c.c. we have
WSTmrAV = ^r':' •! ! y I ,je) 5 ! l0_Scm 
s will give an idea o\ the numerical magnitudes involved
kinetic theory of gases. We give below several constant';
i at a C and atmospheric" 
/,./,■' !
moli
10*
1 city
in
me!
per sec.
Viis
7X10^
\a gm,
cm "'
lee" 1
srmal
conductivity
K X 1C
incaL cm. 1
■src.i "C" 1
318
52
56
339
38.9
Mean
free path
Ax 10*
cm,
Molecu
lar dia
meter
10*
cm.
C!,
43.1
45 2
493
461
1311
413
30?
166
12 I
9.44
9,95
28.5
10.0
4.5"
J. 47
3.5
3.39
218
3.36
4 ■;. ,
taken from Kayc and Laby f T if Physical and
' . fongman. Greet. & Co., 1948. .. at accuracy is claimed
values given.
36
K.JUNE11C THEORY Of MATTER
Book Recommended.
[chap.
A good account of the subject matter of this chapter will be found
in Eugene Bloch, Kinetic Theory of Gases, English translation
published by Me time n and Co.
Other Referen/r .
1. Jeai tic Theory of Gases (1940), Camb. IT P.
2. Kennard, Kinetic Theory of Gta 191 rawHill.
1. 1 [AFTER [V
EQUATIONS OF STATE FOR GASES
1. Deviation from the Perfect Gas Equation.— By the b
Quatiorj oJ State" is meant the mathematical formula which
the relation between pn volume and temperature
of a substance in any state of aggregation. If any two of these
quantiti mown, the third has a fixed value d pending uniquely
o and can be determined if the equation of state is known,
ill is is seldom possible. According to the laws of Boyle and
Charles, we have for a perfect <
pV = RT J)
This is the Equation for a perfect gas.
But even Boyle himself Eound that the law held only under id
ons, viz., high temperatures and low pressure, while unaei
ordinary conditions it did not correctly represent the true state ol
actual gas. Foi every temperature a curve can be
drawn which has for its abscissa the volume and for its ordinate the
responding pressure of the enclosed substance. Tlr ■■. are
led isothermal*, I £ equation (!) wee. isothermals ought
to i ingular hyperbolas parallel to each other, but experime
5 ho n Ehe case, The most extended earlier
.ttions are due to Regnault. He applied pressures up to 30
heres while the temperature was varied from 0° to 100 G. He
duct pV as ordinate ag ;  si a abs< Lssa i ■■■ 6,
7 inira ilii ' I 1 he curves ought to be straight lines parallel
to the xaxis; actually, however, they were inclined to it. He foui
that for air, nitrogen and carbon dioxide the product pV dei
with increasing pressure, while for hydrogen it increases. iU also
found tb n abnormal JouleThomson effect (Chap.
VI). Thesi facts led him to describe hydrogen as "more than
perfect". 1! equation (1) were true the product pV ought, to remain
constant; thus these permanent gases were shown to be imperfect.
Later work by Natterer, Andrews and Cailletet in 'in
sures confirmed the idea that the actual gases showed consid
deviations from equation (1). Andrews' experiments are oJ
fundamental importance as they throw much light on the actual
iviour of gases and form the basis of an important equation of
t'irr proposed by van der Waals. Andrews 1 experiments are
■ibed in the next section.
The most thoroughgoing and exact experiments are due to Amagat
investigated the behaviour of various gases up to a pressure ol
tmospheres. His results particularly" with CO a (Fig, 7) and
1 1 1 . 1 . • 1 1 • showed that their behaviour is very complicated.
A different method has been utilized by. K. Onnes who, investi
i ihe behaviour of several gases at very low temperatures and
that none of the numerous equations of state proposed correctly
us the results of experiments. He finds that at any k ■:■.■
88
i J CIONS OF STATE FOR
[CHAP.
ANDREWS' JIXPERIM)
B9
ture the results are best represented by an empirical equation of
the type
pVstA+Bp+Cp + Dp^., . , . . (2)
e ^, /J, (7,.., are constants for a fixed temperature, buj vary
with temperature in a complicated manner. As many as tw«
constants are used; they are called vinal coefficients. A is simply
equal to RT while the es of the coefficients uf higher terms
diminish rapidly. Holborn and Otto, following Onne* J method,
studied several gases up to 100 atmospheres and between the tempera
tures of 183°C and J 400*C, and found that they need cake only
four cons tits. They give the values of these constants for vai
gases at different temp era Lures.
The coefficient H •, •. particular importance. For all gases h
varies in l simti; way: at low
temperatures it Has a
value, then it gradually in
reases tnd becomes
positive. Now if at; am
peratnre B — ^ and C, D are
negligible as usual, then
dp
Hence, at this a ture'
i.j be obeyed up to
pressures. This tem
rnljrd the
2. Andrews' Experiments. —
While engaged in ttift attempt
to liquefy some of die socalled
permanent gases — an important
problem of those
Andrews,* in 1RG9, was led to
study the isothermals of carbon
dioxide. His apparatus is indi
cated in Fig. 1, aft is a glass
tube whose upper portion con
sists of capillary tube nn.i :'
narrower than the lower part.
Carefully dried carbon dias ide
passed through the tube for
several hours arid then the tube
Fig. 1.— Andrews' apparatus. sealed at both ends. The lower
end of the tube was immersed under mercury and opened, and some
of the gas expelled b 1 heat, so that, on cooling, a small column of
mercury rose in the tube and enclosed the experimental gas.
♦Thomas Andrews £18 'l". ::: :. Born in Belfast, he was Professor oi
Chemistry at Queen's College, Belfast, irom 1845 to 1879. He is remembered
his work in . with the liquefaction of gases.
The tube was surrounded by a strong copper tube A fitted with
brass Hanges at either end, to which brass pieces could be
attached airtight: with the help of rubber washers. A. screw b
; through die lower flange. The tube A contained water
■•■■ n pressures as high as 400 atmospheres could be app
carbon dioxide enclosed above e. To register t ne a
similar rair&Wv tube containing; air was placed on the right siue f m^
in a tube A', exactly similar to A, with which it com
municated through the tube cd and thus the pressures in both tne
tub.es were always kepi equal. The pressure in either tube could be
varied by means of the screws S or S\ The capillary tube a*
be surrounded by any suitable constant temperature bath (not shown;.
3. Discussion of Results The curves obtained by Andrews are
shown in l''ig. 2. Let us consider die isothermal corresponding tc
13. PC. Starting from the
right we see .(portion AB) m>r —
that as we increase the pres
sure, the volume diminishes
laiderably and finally lique
i.ll of the gas begins at a
pressure of about 49 atmos
pheres (point B) , As lon
u Lctifi ' ' nues the
pressure " remains consi.au L
and the volume continually
diminishes, more mid more
of the gas being precipitated
as liquid. This is indicated
by the nearly horizontal line
BC. (The slight inclination
indicating an increase of pres
sure towards the end is due
to the presence of air as im
purity) . At C all the gas has
condensed into liquid and the
almost vertical rise: of the
curve indicated by (JD corres
i ds to die fact that liquids,
nly slightly compressible.
The isothermal corresponding to 21.5°C is of the same gen
form but the horizontal portion B'C h shorter. In this case the
icific volume of the vapour when condensation begins aller
le that of the liquid when condensation has completed
mi die corresponding volumes lor the previous curve. As
tperaturc is raised these" changes proceed in the same direction
as above, till at 3L1~'C the horizontal pari 1,  just disappeared and
Specific volume in <■£
Fig . nnak for Carbon d
ailed the critical
two volumes have become identical. This \s
isothermal for carbon dioxide. Above this temperature, the horizontal
part is absent from all the isothermals and as we increase the: pressure
90
EQUATIONS OF STATE FOR CASES
CHAP.
SSdlv  riTi ■ i " formatI ? n eL&lMi<L but ,,u: volume diminishes
it becomes nearly equal to , ■•!„;„, , ., , ]iqui j at a
Ink lower tcnmey alure . This peculiar! i, o\ ffi isothermal alio ■
disappears at higher temperatures as is evident from the isothermal
for 48,1 °C which is much like the isotherm als for air shown separately
on the righthand top.
We thus see that the whole diagram for carbon dioxide is divided
iiv the critical isothermal into two essentially different regions.
Above this isothermal nc liquid state is at ail possible even under
the greatest pressure, while below k there are three separate regions.
In the region enclosed by the dotted curve BB'PC'C whose highest
point P, railed the critical point lies on the critical isothermal, both
lid and gaseous states coexist. To the left of the line PC and
below the critical isothermal there is the liquid region while to the
■ '. PB there is the gaseous region. Now if, by means of gradual
changes, we want to convert gaseous CO.. at 25 C and 60 atm. pres
sure {represented by the point R) into liquid CO. at the same tem
perature (represented by the point S) without any discontinue
appearing, i.e., the mass is not to separate into a liquid and a gaseous
part with a layer between them, we must avoid reaching the inside of
the dotted curve BB'PC'C. Thus we heat the substance above: S3 i
and then compress il till the volume he< i to thai of the
liquid at that temperature. Next coo] >°C and then reduce the
sure. Thus starting from the point r which undoubtedly repre
o/ ike liquik and easeo ,d was
[merits of Andre, .
bon dioxide is compressed above
•i.i f no liquid can make its appearance, however rem the pressun
More accurate experiments show tfrtfc this temperature ; ;
31.0* and not 3T.1 . The temperature at SI J i is ca led the e
(T t i for carbon dioxide. We may define critical tern .
ture as the highest temperature at which a gas can be liquefied 1
pressure alone. This is why the earlier attempts to liquefy the
permanent gases failed, though enormous pressures amounting to as
hi:,! as b,000 atmospheres were sometimes employed. Tli pressure
necessary to liquefy gases aL the critical term is called
critical, pressure (p t ) and the volume which the gas* then occupies
at the critical temperature and critical pressure is called the
me (V,) These three quantities are called the critical
mtt of a gas. A table giving the critical constants for various
gases, S given on p, 102, It obviously follows that in trying to
iquej, . It uMdw to apply pressure alone if the initial ,
v : :i :: ;; .. 'v : ' ■ '*■ "* «< ■>,.•, ■■■, ,;, ;1 , M ,,, lhr crisi „,
VAN DEE WAALS' EQUATION
91
4 Van der Waals' Equation of State.— 1 he equation of state for
i gases was deduced theoretically fiom the kinetic theory of
(Chap. Ill) . There we assumed that the molecules have no
volume and do not. exert forces of attraction on one another These
assumptions are correct only for the ideal or perfect gas to which the
actual gases of nature approximate at low pressure and high tempera
ture. Rigorously speaking, however, all gas molecules nave Imite size
whose importance was first pointed out by Clausius. tor our pr.
purpose we shall treat the molecules as hard elastic spheres. It is
clear that at very high pressures the total volume of all the molecules
will not be a negligible fraction of the volume of the gas and further
even at the highest possible pressure the volume occupied by the
substance cannot be less than the volume occupied by the molecules
when they are most closely packed. It follows, therefore, that the
free volume of the gas to which the BoyleCharles' law refers is not
the *ras volume V but is less than V by a factor b where b is related
the total volume of the gas moleci
Another simple way of arriving at this result is from considera
tions' of collisions. Consider four balls lying separated from one
another on a line perpendicular to the wall and let the farthest one
start to move towards the wall with a fixed velocity, if the balls are
big, die distance to be traversed is less and hence the last, ball will
strike the wall earlier than when the balls are mere points. Similarly
for molecules of finite size the number of collisions with the walls
and hence the pressure will be greater than for point molecules. Thus
the effect of molecular size is equivalent to a reduction in the total
volume of the gas by b.
Hirn, in 1864, pointed out that the molecules must exert forces
of attraction ov one another, hence the energy cannoL be wholly
kinetic, and potential energy due to forces of cohesion must ah
taken into account. The correction for Forces of cohesion can be very
simply obtained. These fortes are of the same nature as those which
give rise to the phenomenon of surface tension in liquids. The mole
cules attract one another with a force which varies ly as some
power of the distance between them. Thus die force will be appre
ciable only for small distances and is negligible for larger on: ■•=.
molecule in the interior is acted on by forces in all direction's and hence
these will balance ; but this is not so with a molecule on the surface
or close to it. The components of the forces acting on it resolved
parallel to the surface wiM balance but not. those in a perpendicular
direction. There will be a resultant force acting perpendicular to the
boundary layer and directed inwards. It is obvious that this force on
■ le molecule will be proportional to the number of attracting
particles in the fluid, i.e., to the number n of molecules per c.c. The
force acting' per unit area of the gaseous boundary will be proportional
to the product of the above force and the number of molecules in that
area. Hence the cohesive force /?, acting per unit area of the boundary
layer of the gas is proportional to ?t~. Now n = N/V where N is the
total number of molecules and V the total volume. Therefore
/>, oc l/V 2 . This force opposes the outward motion of the molecules
92
EQUATIONS OF STATE FOR GASES
[chap.
aiifl thus decreases their momentum and hence also the pressure
generated by their impact. Hence the pressure will be less than that
calculated previously by the factor a/pa, where a is .some constant.
In other words, we must replace p in die perfect gas equation b}
u
P + p 1,s 'C the external pressure p on the gas is increased by a/V B .
Applying both these corrections the gas equation becomes
[*+£)[v*)*T. . . . .
This is van dcr Waals' equation*. Detailed considerations show
that b is equal to four times the total vol tune of the molecules. Van
der Waalsf was the flrst to work out a systematic theory, taking into
account both these factors, viz., the finite size of molecules and the
ces of cohesion. Van der Waals' equation is found to hold n
;i deviations from the perfect gas equation. A comparison of
tins with the experimental results is given in sec. 6. Various other
equations of state have been proposed. Some of them are more
accurate than van der Waals 1 equation in certain regions ; nevertheless,
latter, considering its simplicity, gives in general, the most
satisfactory first approximation to the behaviour of actual eases. We
shall discuss this eqj , re detail.
5. Method of finding the Values of V and V— \ method of
 of W and <*> ocairring in van der Waals' equation
1 5 given below : — '
rithal Data,— In the next
n from theoretical considerations that the
■e T„ critical p: l( and critical volume
i obeying van der Wa > , vely givt:y
• ■ (5)
■ (6)
1 c 27!»R ;
Pi H p c
Thus j I T * are experimentally determined f a' and 7/ can be
I wit* the help of equation (6). It may, however" be
pointed out that the method is not very achate u the \as do t
no obey van der Waals' law accurately near the critical pomt. But
ot of much consequence as V and <b' themselves defend
upon temperature and volume. The values of V at d ' >' Lffev
miportanl gases are given in table 1. They refer to 3 rcfof
:u. X. I .K, and are determined by this method from equ I ^
on hJS \t ^'a^'f ° f **"** thls «»«*■ "» ** Author,' Treaty
tJohatmes Diderik ran der Waals (18371923) was born at Lcvdeu m U&t
1)3 Prize in 1910 ^ *"*"* « ^T**** and was Warded
< r W)1S^ : ^t ! ^Jr,^ »*** Y !a the table by
D S< , ssi ON OF VAN DTTJl WAALS EQUATION
' ff » 7. — j:\v./r."'.x of 'a' and 7/ /or some gases.
.
Substance
a X W«
in atm. )< cm. 4
& x 10*
in ex.
ium
6.8
106
Argon
2m
1 X'
Oxygen
m
i
Nitrogen . 
I 27:1
lV:i
Hydrogen
48.7
US
Carbon dioa
71V
•
Ammonia
8BS
1GG
■nes x cm, 4 = 3 5 x LO 4
. , ise.— Using the values of T e = 5.3, p e — 2.HS atm., calcu
late 'a' and 'b' for helium tor a grammolecule,
27JP 7 ! ■ x:a3x ]Q 7 ) a xJ53)a
■'  64 " ,*, ~ 64x225x1 '01x10 s
atm. x cm, 4
5 3x8 ; 3xl0 7
h
,8xl01xiO e
Discussion of van der
equation. — We shall
van der Waals'
=24 e.e.
6.
Waals'
now disi
iion
(/>+;; x « r 
This is an equation of the
.] degree in V, hence it
ows that for every Kalue
of p, V must have three
values. Further, iroin theory
of equations, either all the
three values are "real or one
is real and two imaginary.
Again writing the equa
tion in the form
RT a , n .
P = yZTl >yT * (')
see that for very large
values of V, p is small and
in the limit /;=0 when F=Ga.
Any in, when V is very small
approaching b, p tends to in
finity. Hence the curve must
have a concavity upwards.
Further V cannot be less than
g
X ■;,,,
i
Theoretical curres
for carbon divxidz
Specific volume
Fig. 3— Theoretical curves for CO= accord
ing to van dcr Waals' Equation;,
* Taken from Landolt and BurDSlcin, PkysikaliscfeChemische TabcHcv.
94
EQUATIONS OF SI ATE FOR GASKS'
ns.
— .*.«— ««g equation
by means of graphs lor every temperature (p = ordinate, V = abscissa)
curves of die type shown in Fig, 3 are obtained. It is readily seen that
tlie turves resemble, in general, the experimental curves (Fig. 2)
obtained by Andrews, but if we look for quantitative agreement bv
trying to make the two sets of curves coincide we are greatly dis
appointed. In fact, the agreement is only approximate and qualitative
party because of errors in the assumed values of V and 'b' and
partly, because the equation holds only approximately.
There is, however, a remarkable divergence between the theoj
iiLiu and the experimental curves in one region. The theoretical
curves drawn from van der Waals' equation give maxima and minima
in me region represented by straight lines in Andrews' curves. Expert
mentally, this u the region where condensation or vaporisation begi
and die pressure remain, constant as Ion as the process continues.
1 his > difference is easily explained when the theoretical curve is.
properly interpreted. The part bd inside the dotted curve correspo
to the fact thai the volume should decrease with decrease of pre,
which a quite contrary to experience. This would be a collapsible
state, for any decrease of volume is accompanied bv a decrease of
pressure which tends to further decrease the volume. (This is
>arent it we imagine the fluid to be confined in a cylinder ') Thus
state of the fluid inside the dotted curve represents a state of
and consequently, can never he realised in
This is why the part bet is not obtained in Andrews'
ion ab represents supersaturated vapour and is
med experimentally as, Tor exam hen air con
ater vapour is compressed beyond the pofm when condensa
iir, without conden^tion occurring. This
■ from dust or charged ions which ac«
T This state is, however/ m,,ableandteasik
d by the introduction of particles of dust, etc. Hence the
portion i M represents supersaturated vapour in unstable equili
brium does not occur ,n Andrews' curves which represent oniylZL
of .table exmihbrium. Similarly, the portion de represents a supS
hen ted hqmd which is also in unstable equilibrium and is obtained
experimentally when gasfree liquid is carefully heated Hcmce is
^nce d re^LT r m An4 ^ ~ Thl ** W^diver!
Van der Waals' theory, however, does not tell us when conden
sation begins, ,.*., where the straight part commences. A Imp e
thermodynamic argument* shows that die straight portion shoKe
trawn i bat the area abca = area cdec, ' l
mi
EQUATION
95
maxima and minima points on the theoretical curve can be
I b\ putting .^0. Hence from (7) , by differentiation,
dp RT 2a _ Q
< 
(8)
(9)
2a(Vb) s
RV' A '
i ; cubic equation. Hence, for every isothermal there are
i ir one real and two imaginary [joints of maxima or minima.
I In uii vi s below P in Fig. 3 are seen to possess one maxima and one
n :i point while those above it have none at all. A slight mathe
nsiormation will show that the other point of minima lies.
ion f '<■!.' and hence has no physical meaning. Equation (7)
mi. I i'.i when combined yield
.' <V •
P ~ ~ pa
;io)
i ih.; i urve passing' through the maxima and minima points and.
n by the clotted curve QPR.
In Fig. 2 all isothernials lower than P cut the dotted curve
at two points and, therefore, liquefaction can be observed
nging the volume along them. For the critical isothermal,
1 1 rraal corresponding to the critical temperature, these
His have coah nto one. Referring to Fig. 3, it is readily
iai the isothermal passing through the point P, where P is a
! in! lej ion for the family, of curves or a maxima point for the
urve, is the critical isothermal because below P every isother
got maxima and minima points while above P there are
I all. As we have seen above, if the isothermal has g
i. ; ■ linima points there must be liquefaction of the ;
i n< I we can find out the position where liquefaction begins. Foi
e must be two points on the curve having equal
ure. I he highest 'isothermal for which this condition is satisfied
tl issitig through die point P since at P the maxima and
i points have c >a U seed into one. Hence P in Fig. 3 must be
Identified with the critical point and the isothermal through it with
tin critical isothermal. Now for p to be the maxima point of the
1 : have, by differentiating (19) with respect to T r and
pqu i in to zero.
a_ Sa(VrZb)
i (Ifl'i
mm! I ' ■mi ■
f/4
V t = 34.
b — a
P* • 2"b z
I
= 0,
27m
96
EQUATIONS OF STATE FOR GAS]
[chap,
r ri? e ^f C0 T nSta/ be ver y easil V ded «ced from (7) since
hi the cntrcal isoiherma] ft* point P is a maximum poin't £ \ ' ,
a point of inflexion, and for it both § and *£ *? e equal to „,.
' have therefore
dp _ _ #7* 2n n
• •")
2)
•iRIM.F.NlAL STUDY OF EQUATION OF STATE
97
=0,
^bluing (8) and (12) we get p f = 5*, and then with die
:,nd %£L=*
, 3 '
7. Defect in van der Waals' EquaUon.fn spite of
agreement deductions from van der Waal* equation show
►le deviation from experimental result,. 1 We haVe r? •
shown that for carbon dioxide the curves drawn I
SfP. f "■;■' ^JMkeexp.nmen^
fain van der Waals' equation *ivp$ w — a*, ,..t.i
: II h found tha. p, i, Sr^equ^o 2* W "P""" 8 "
I
Compression
pamp
[ he apparatus is shown in Fig. 4. T is the measuring vessel
lass which ends in a
M h calibrated capillary tube.
urt of the vessel is placed
iteel cylinder. By means of a
V and a screw E the
! is held in the steel cylinder.
mostat surrounding the
i rt of the tube which
losed by means of a mantle
.il T is filled with the gas
i investigation at atmospheric
the space between T and
the inner wall 11 being filled with a
iiLn ii. in: quantity of mercury over
ch a quantity of glycerine or
in oil is poured. Pressure is
licated from a compression
1 1 through glycerine to the
miiig vessel. At high pressures
tner< ury will rise up to the capillary
' in and the volume can be easily
i r from the calibration. When
iIm pressure is high, it is measured
by a compression manometer. If the
is above 300 atm, and is
ted only from the inside the capil
ii l generally smashed, hence Kfc 4.— CajHetct'a apparatus.
ih' experimental tube should be subjected to pressure from all sides.
uih ;i pressure tube was built by Amagat.
Amairat carried out an exhaustive study of
the behaviour of several gases by the above
method. In one set of experiments he employed
pressures up to 450 atm., while in the next series
pressures as high as 3,000 atmospheres were
used.
(b) Apparatus based on the principle of
x'anable mass, — This method was employed by
Holborn and Schultze. Hoi born and Otto and
Kamerlingh Onnes. These investigators worked
at high pressures and obtained important results.
Since with increasing pressure the volume be
comes smaller, greater error would occur in read
ing the volume at high pressures. To avoid it,
these investigators kept the volume constant and
used different quantities of the gas whose masses
were determined. The apparatus becomes some
what corn.pl i rated by the presence of devices for
ntroduction and removal of the gas both inside and outside the
7
Fig. S. — The
1 1 : sure balance.
98
EQUATIONS OF STATE FOR CASES
[ CHAP*
experimental vessel An ingenious pressure balance shown in Fig, 5
was used to measure such high pressures. The metal block B was
firmly damped in position and carried the tube T ^frich was con
nected to the apparatus containing the experimental gas. The block
B has also a cylindrical hole in which the cylindrical rod R accurately
fitted. Between R and the gas in the tube T there was castor oil so
that gas pressure, transmitted through the oil, tended to raise the
piston R. This was just prevented by the screw S pressing on the
top of R with the combined weight, of the frame F and weights W,
When balance is obtained, i.e., the piston neither rises nor falls, the
gas pressure p z=mgfa, where m is the mass of R, S, F, and "W, and
a is the crosssec Lion of the piston or the cylinder.
9* Discussion of Results.— Amagat represented his results by
graphs in which pV de
notes the ordinate and p
the abscissa. The curves
for hydrogen and nitrogen
for several temperatures
hown in Fig. 6. As
already mentioned, for hy
drogen tire product pi' in
creases with pressure, but
for nitrogen it first de
creases. The curves are
straight lines inclined to
the pressure axis, while if
Boyle's law were true,
they would be straight
lines parallel to the pres
sure axis. The curves for
carbon dioxide (Fig. 7) are typical of all gases. The Isothermals
50° have a portion of them parallel to the jbFaxis. This
indicates that the pressure remains constant while the volume varies,
and corresponds to the condensation of the vapour. Further, it is
seen that the curvature of the isothermals diminishes as the tempera
ture rises. The minima points Cm isothermals gradually recede away
from the origin, and the dotted curve through them is parabolic. At
still higher temperatures no minima point is found and carbon dioxide
behaves like hydrogen.
This general behaviour of tile isothermals can be easily explained
from van der Waals' equation. We have
j.OOO S.OOO
p In atmos.
s,ooo
6 — Amagat's curves (FV again;;
for different gases.
m
In the third and fourth terms which are small, we can make the
approximate substitution V = RT/p. Eqn. (IS) then yields
SSION OF RESULTS
**+&&.**■
99
(14)
emperatures the term abp' 2 /K 2 T 2 can be neglected. If we
then ] us ycoordinate and p as .^coordinate, the plot will be a
inclined to the pressure axis (e.g. curves for H 2 in
too ia& lg3
p in atmos.
Fig. 7. — Curves for carbon dioxide.
["bus for temperatures above the Boyle point T B = a/bR
will always be positive as for II 2 in the figure. For tem
peratures below the Boyle point as in the case of N 2 and C0 2 in
ii and 7, the slope will he negative unless the pressure is too
' • 1 • 1 1 . This can be readily seen from eqn. (14) which holds for the
al case, since
slope = £^' + J*. . (15)
ilope is therefore negative at low pressures but becomes positive
m ii iently high pressures. The minima point on the Amagat curves
i<\, the point where the slope changes its sign, can be obtained by
•qua ting (15) to zero. Thus corresponding to any single value of
1 are two temperatures given by the relation
• ■ (16)
b ~w
= 0,
R*T*
lotted curve through the minima points is approximately
i' 11 1 in Eqn (16) shows that the dotted curve will meet the axis
in she Boyle temperature T B = afbR. Thus Amagat's curves
In ltd \ ' •:. ilained with the help of van der Waals' equation.
too
EQUATIONS OF STATE FOR G
[QIAy
10, Experimental Determination of Critical Constants.— We have de
fined the critical constants in sec. $. They are constants characteristic
at every substance, and are oi' fundamental importance as they occur
in certain equations of state. Their importance in the study of
liquefaction is discussed in Chapter VI.
The determination of these critical values is often a task of
considerable difficulty. Of these Lhe critical temperature is the easiest
to measure accurately. For ordinary substances* a hard glass tube
like that of Andrews and connected to a manometer may be employed.
Sufficient quantity of the liquid is introduced and the tube surrounded
by a thermostat which can be maintained at constant temperatures
ring by very small amounts. The temperatures at which the
liquid suddenly disappears and reappears are observed, the mean of
these giving the critical temperature. The critical pressure is the
pressure at the critical temperature and can be read easily from the
manometer. The critical volume is much more difficult to measure
accurately, for even a small variation of temperature by 01° C pro
duces a large change in volume, and hence the substance has to be
kept exactly at the critical temperature. The pressure must also be
exactly equal to the critical pressure since the compressibility of the
substance in this region is very great. The method adopted was to
arrange in such a way that a very slight increase of volume low
the temperature by a small amount and caused the separation of the
into liquid and vapour, the liquid appearing at the top. This
initial volume is called the critical volume. The amount of substance
initially contained in the tube has thus to be adjusted.
The most accurate method,
is to make use of the
Law of Rectilinear Diameters
or mean densities, disco
by Cailletet and Mathias. If
the density of a liquid and of
its saturated vapour be repre
sented by ordi nates and the
corresponding 1 temperatures by
abscissae, a curve roughly para
bolic in shape 5s obtained (Fig.
8). In the figure the vapour
density of nitrogen is plotted
from the observations of On
nes and Crommelin and is
densities in the two states go on
Temperature '. itigradeu
Fig. 8.— Law of Rectilineal" Diameter
typical of all substances. The
approaching" each other till they become equal at the critical tempera
mctimes a simple apparatus, first suggested by Caffniard do la Tour is
yea* for the purpose. Il consists of a glass tube shaped like T, w
arm somewhat broadened and containing the liquid :n question which is
l air in the larger arm by a column of mercury. The two end
closed, the air in the longer arm serving as a compression manometer. The
■" :.r,ir:cc of the surface of separation between id and its vapour in
the sli >rter arm was observed.
The curve AB is a line passing through the mean of the vapom
i quid densities and will consequently pass through the critical
Lempen •. It was first observed by Cailletet and Mathias that for
this line was straight or very nearly so. The equation
; y = %(p r \p n ) = a[bt where y is the ordinate and t
■ in abscissa and p u p v denote the densities in the liquid and the
lit; respectively. This law enables us to find the critical
■ the critical volume, for we determine the densities of the
..•apoii and liquid as near to the critical temperature
hen draw the rectilinear diameter. The intersect!
ivith the ordinate at the critical temperature gives lhe critical
iy p c or the critical volume.
For substances like water which attack glass at high tempera
la illetet and Colardeau employed the apparatus shown in
strong steel tube AB, platinized inside to prevent attack,
hi ins the water "or the substance to be investigated. It is immersed
im ,i temperature hath LL which is heated by a gas regulated burner,
l
Fig. 9. — Cailletet and Colardeaifs apparatus.
temperature can be kept constant. The tube AB is
a similar steel tube FG by means of the flexible steel
Mercury fills part of the lube AB, the spiral CDE
he tube FG up to the level $ lt above which there is water filling
tube up to the manometer. Different pressures can be
i! 'I 'I i I by the force pump as indicated, At S t an insulated platinum
h is sealed in the side of the wall completes an electric bell
n the temperature of the bath is raised the pressure of
hi. I thus its
ted to
ipiia] CDE.
EQUATIONS OF STATE FOR GASES
[ CHAP.
2 *0
the vapour in OB rises, mercury is forced past Sj in FG and sets
the bell ringing. Water is forced in by the pump to keep the level
of mercury constantly at S x and thus Uie volume occupied by the
water and' water vapour in AE remains constant. The platinum
vrirc at Sj completes another electric circuit and serves lo sound a
warning that die whole mercury is about to be expelled out of AB.
We thus get the vapour pressure curve of the substance. The
curve is perfectly continuous and characteristic of the substance.
for water this is shown in Fig.
10. If* however, we start with
different quantities of the liquid
we get the same curve as far as
M, but above it we get different
curves. The vapour pressure
thus appears to branch off at M,
a point, whose position was
found by Cailletet to be practi
cally independent of the quan
tity 'of liquid taken. This tempe
rature is the critical temperature
for the substance,
11, We give below a table
of critical constants, taken from
Landolt and Bernstein's Physi
cal Chemisch e Tab el ten.
300 320 3iO 360 3&0 4 00
Temperature centigrade.
Fig. 10. Vapour Pressure
Table Z— Critical data.
T c
P<
Bi
RT t
Gas
in °C.
in atm.
Specific
Pc V*
225
volume
Helium
2679
154
313
Hydrogen
2309
12S
;v\ 2
328
Argon
1229
480
188
343
Oxygen
1888
497
2S2
342
Nitrogen
— 1471
33 5
321
S42
Carbon dioxide
SI 0
728
217
318
Ammonia
1322
1123
424
412
Ether
1938
356
385
3 • B 1
Sulphur dioxide . 
1572
776
195
3 CO
Methvl chloride
1481
659
271
380
Water
3742
220
26
12. Matter near the Critical Point— There has been much dis
cussion about the slate of matter near the critical point since the
time of Andrews. The properties actually observed are: — • (1) the
MATTER NEAR TUF. CRITICAL POINT
10!
and hence there must be mutual diffusion, and the surface tension
must vanish, i.e., the molecular attraction in the liquid and vapour
slates must, become equal ; (3) the whole mass presents a very flicker
appearance which suggests that there intent be variations of
densitv inside the mass. This was experimentally observed to be so
by Hem and others. They suspended spheres of different densities
inside the fluid when each comes to rest at a horizontal surface
Saving a density equal to its own; (4) compressibility of vapour at
rhe critical point is infinite and is very great near that point. As
pointed out "by Guoy, this explains the variation of density through
out the mass observed in (3) , for the superincumbent vapour causes
the density of lower parts to increase.
From these considerations the simplest and probably the most
correct view which was put forward by Andrews appears to be that
just beyond the critical temperature the whole mass is converted into
vapour' consisting of a single constituent and should behave like a
gas near its point of liquefaction.
According to this theory the critical phenomenon, i.e.., the dis
appearance of the boundary between Che liquid and the vapour and
not its motion, should occur only when the amount of liquid in the
tube is such that it will fill the whole tube with vapour of critical
density. If more liquid is present the meniscus should go on rising
till at the critical temperature the whole tube becomes filled with
id. If less liquid is present, the meniscus goes on falling till
at the critical temperature the whole should become filled with vapour
alone. Experimentally, however, Hein found that the critical pheno
menon is observable when the initial density varies from 0735 to
1 269 times the critical density. This is probably due to the property
(4) as the variation of density inside the mass allows the excess or
deficit amount to be adjusted. ' The branching of the vapour pressure
curve at M observed by Cailletet and Colardeau may be explained
in a similar way.
Experiments with water by Callendar point to the existence of
a critical region rather than a critical point He found that the
ity of the liquid and the vapour did not become equal at the
temperature at which the meniscus disappeared, but that a difference
of density was perceptible even beyond that temperature. The criti
i 1 1 point is that point at which 'the properties in the two phases
line equal.
Books Recommended.
1. Jeans, Kinetic Theory of Gates, C. U. P. (1940).
Kennard, Kinetic Theory of Gases, (1938),
. Glazebrook, A Dictionary of Applied Physics, Vol. I.
•
CHAPTER V
CHANGE OF STATE
Fusion — Vaporisation — Sublimation
1. It is a matter of common experience that on the application
of heat, »ub&tances change their state o£ aggregation. Thus when ice
is heated it melts into liquid (water) at 0°C (the melting point) .
tion point and later the liquid solidifies at the freezing point or
solidification point. For a pure substance the melting and the freez
ing points are identical, as are also the boiling and the liquefaction
points.
The temperature at which any change of state takes place is
generally fixed provided the external pressure is fixed and the subs
lance is pure. The fusion point usually varies very little with the
pressure (it requires a pressure of about 130 a tin. to lower the melting
point of ice to — 1°C), but the variation of the boiling point with
pressure is very great. As a matter of fact, it can be easily shown
water can be made to boil at any temperature up to 0'"C.* pro
vided the sufficiently reduced. Conversely, the boiling
iderably if the pressure be sufficiently
These [acts can be readily verified by considering evapora*
sed space.
% Evaporation in a Closed Space.— If we fill a glass vessel partly
with water and evacuate it with a pump, then water will begin
to boil even at room temperature. If the pump be now cut off, the
pressure can be measured by a manometer. For a certain definite
temperature of the liquid, there is always a definite vapour pressure.
Jf we increase the total space, more liquid will evaporate and fill up
die extra space. If we reduce the space, .some vapour will condense
till the remaining vapour exerts the same pressure. If there is a
third gas., noL reacting with the vapour, the partial pressure of the
liquid will be approximately equal to the vapour pressure in the
absence of the third gas. It. is an important experimental task in
Physics to determine "the vapour pressure of a liquid at different
temperatures.
3, Latent Heat. — Black found that the change from one state
to another is not abrupt, but a large amount of heat must be
absorbed before the entire mass is converted from one slate to
another at the same temperature.^ Thus to convert 1 gram of
* More rigorously, up fcr> the triple point.
t Wc have already discussed in Chap. II, pp. 3337, the methods of measur
ing quantity of heat h> Change of State
»■]
CftANGE OF PROPERTIES ON MELTING
L05
ice at 0°C to 1 gram of
of water at 0°C. about 80 calories of b
red The amount of beat required to convert 1 1 gram ot a
folidlntoTliquid without raising the temperature ,. called the fa*
ioua ™* , .**"i TW amount of heat is required for overcoming the
,, ,, lC mobile enouch to form a liquid. In solids the molecules
^?ima £ne< Ta? "bralin about mean equilibrium positions which
•re fi«d but in liquids they execute rotational ami trans at onal
motionflnd wanderVou^hout the liquid, Oiough con»derably *
nered Similarly, to convert one gram of water at 100 C to vapour
^1 0C 5387 calories are required. This heat which is neces
sary or pulling the molecules of water so far apart that they beo „
qute ndependent of each other (vapour state) r, known as t.
Stmt heat of vaporisation. It will be seen that the latent heal varies
greatly with the temperature of vaporisation.
A Sublimation, Sometimes a solid may pass to a vapour state
withLSSh^rfirough the intermediate liquid state. Camphor
rrnrles a good example of this class. On being heated it does not
;r. but sSnply evaporates. Such a process is died subhmcUon
and the substance is said to be volatile.
But we shall see that the process is not peculiar to am parti
All solid substances possess finite vapour tension
When this vapour tension
cular substance.
at even ordinary temperature
small we take no notice of it, but with the aid o delicate apparatus,
ii ran be measured, A substance is said to be volatile only when the
boiling point at atmospheric pressure is less than the melting point.
Thus under an atmosphere different from our own, say at the moon.
even ice which we do not consider volatile would have to be treated
as such. The moon is supposed to have a very thin atmosphere
t <1 mm. of mercury), and the temperature is below C. If
Consider ourselves transported to die moon, our studies will show
that ice h volatile because on being healed, it will evaporate to the
gaseous state without passing through the liquid state. It we wain
water in the moon we must artificially produce a high pressure
and applv heat to ice under this pressure. Similarly, camphor can
be melted to a liquid form when it is heated under high pressure,
5, Amorphous Solids,— Impure substances, mixtures, and non
■stalling subsLances do not usually have a sharp melting pint;
in' their case fusion and solidification take place over a short range
of temperature. This is due to the presence of two or more sub
stances which do not solidify at die same temperature^ Examples oi
hous substances are wax, pitch, glass etc. Glass gradually
tens throughout its bulk as its temperature is raised and is usually
irded as a supercooled liquid.
6, Change of Properties on Melting.— Several properties of sub
i ces change in a very marked way when a substance melts. Ous
Lo the regular arrangement of' the molecules in the solid being
■ed by the addition of heat. The following are some of these
fin iperties :— 
106
l HANOE OF STATE V
[ CHAP.
(1) Change of Volume, — Most substances expand on solidifica
tion while a few others contract. To the former class belong ice>
iron, bismuth, antimony, etc.; paraffin wax and most metals bi
to the latter class. Good castings can only be made from substances
of the former class. Often enormous force is exerted by water when
it freezes into ice. The burs ting of water pipes and of plant cells
and the splitting of rocks is due lo this cause,
(2) Change of Vapour Pressure. — The vapour pressure abruptly
changes at the melting point. The vapour pressure curves of the
solid and liquid slates are different and there is a sharp discontinuity
at the melting point (Chap, X, § 41).
(3) Change of Electrical Resistance* — The electrical resistance
of metals undergoes a sudden change on melting. When the sub
stance contracts on melting the conductivity increases and when it
expands on melting the conductivity decreases. Table 1* below gives
the ratio of the resistance of the fluid metal to that of the solid form
at the melting point for a number of metals.
Ratio
Table 1.
resistance of fluid meted
rests to n cr o } crystallised metal
for some
Substan
Ratio
Substance
Cu
Ratio
Al
197
070
Li
196
20
Na
134
Cd
197
Ag
T9R
Cs
>:;.
f'.i
0 ■!:■
Ga
058
Zn
200
An
228
Sn
2 
K
132
(4) It is also found that molten metals show a discontinuity
in their dissolving power at the melting point,
7. Determination of the Melting Point. — The melting point under
normal conditions can be determined with very simple apparatus.
The substance may be heated In a crucible electrically or otherwise.
For high temperatures, the crucible must be of graphite or some other
suitable material, and the substance heated in a nonoxidizing atmos
phere. If the substance is rare, it can be employed In the form
of a wire (wiremethod) . As thermometer, the secondary standards
are very convenient to use, the thermocouple or the resistance
* Taken from Handbwh for Fhysik, Vol, X, p, , : 7.
DETERMINATION OF THE LATENT HEAT OF FUSION
107
thermometer being generally chosen ; such th ermocouples or resistance
wires must not be thrust directly into
the melting substance, but should be
protected by a sheath of protecting
material say porcelain, hard glass or
magnesia tubes. Now as long as the
substance is melting the temperature
remains constant and hence the E.M.I?.
of the thermocouple or the resistance
the thermometer is also stationary.
curve is plotted with the E.M.F.
the resistance as ordinate and time
abscissa. The horizontal part re
presents the freezing of the metal and
rhp rrvnpsnondintf EM.F. gives its
of
A
or
as
the
Fig.
1.— Melting point of
copper.
is
about 105 mV, whirl;
corresponding
melting point. Such a curve for copper
is shown in Fig. 1 where SQPMORh, Pt
couple is used. The constant E.M.F.
corresponding to the horizontal part
corresponds to 1084°C,
8. Determination of the Latent Heat of Fusion. For determining
the latent heat of fusion, an ordinary calorimetric method (Chap, H)
may be employed, e.g.^(l) *e *>«** of ^^ <?. ™ *?"?
ice calorimeter, (3) the method of cooling, and (4) electrical methods.
(3) is unimportant, and will not he considered here.
The method of mixtures.— It is quite simple and has been ex
plained in Chap. II. Most of the early determinations of the latent
heat of ice were made by this method. Thus if M grams of ice at
0°C arc added to a calorimeter containing water whose total thermal
capacity is W and initial temperature 0i and if <? a be the final tem
perature, the latent heat L is given by the relation
ATI+Mff^H^flJ. .... (0
An important source of error in the above method lies in the fact
that some water adheres to the crystals of ice at 0°C. To eliminate
this ice below C C is frequently chosen which requires a knowledge
of the specific heat of ice. In this method we are required to find the
t taken up by ice in being heated. A converse method may also
be employed, viz., the heat given out by water in solidification may
found. The most accurate experiments give the value L =zl*db
i ;if. for the latent heat of fusion of ice.
The method of mixtures has been very conveniently adopted to
the simultaneous determination of the melting point and the latent
heat of fusion of metals and their salts. Goodwin and Kalmus ern
this method for finding the latent heat of fusion of various
Salts, A known weight of the substance contained in a sealed plati
m vessel is heated' in an electric furnace to a high accurately
i. limbic temperature. It is then dropped into a calorimeter and
iiiiLv of heat liberated is determined in the usual manner. The
108
CHANGE OF STATE
[ CHAP
2.— Latent Heat oi
Fusion of Salts,
electric furnace is specially designed to secure a uniform temperature
throughout the platinum cylinder which is measured by a PtRh
thermocouple. As calorimctric liquid, water was employed below
450 °G and aniline above that, temperature.
First a blank experiment gives
the heat capacity of the platinum
vessel. The experiment is then
performed with the substance
heated to different initial tempera
tures extending over a range of
about 50 °C both above and below
the melting point, and the final
temperature of the calorimeter
noted. From these after correcting
'* c for the heat capacity of the vessel,
the quantity of heat Q necessary
to raise 1 gram of the substance
from the room temperature to its
in ial temperature could be calculated. Plotting Q_ as ordinate and
T the corresponding initial temperature as abscissa curves of the type
•shown In Fig, 2 are obtained. The discontinuity in the value of Q
indicated by the vertical line gives the latent heat of fusion, the tem
perature at which this discontinuity appears is the melting point T m ,'
and thi it any temperature gives'lhe specific heat
be substance at that temperature.
This methoi rably improved by Awbcry and F.
iffiths who have made acci : : termination of the latent heat of
iveral metals. employed a very special type of calori
tei which d that the heated substance could be kept
surrounded :r inside the calorimeter and the lid of the latter
rater hud access to it, so that the loss of liquid
by e iti n ed.
ctrical Method. — This consists in measuring the amount of
electrical energy required to heat a mass of the substance below its
melt nt to a temperature above it. The method was employed
by Dickinson, Harper and Osborne Cor finding the latent heat of
i of ice. Electrical energy was supplied to a special calorimeter
similar to that of Nernst in which ice below D C was placed. Then
E
 J n
c pi dT+L
T l n
Ct,«dT t
where E denotes the electrical energy supplied per gram. 7*^, T 3 its
initial and final temperatures and c^ c p ^ the specific beats of ice
and water respectively. From this relation L can be calculated.
This mediod is very convenient for finding the latent heat at low
temperatm
Methods similar to the foregoing can also be employed for folding
the heat of transformation of one allotropic modification to another
•bur we are not concerned with them here.
v.]
EFFECT OF PRESSURE ON MFLTINC POIN 1
109
9. Indirect Method.— Another method of finding the latent heat
sts in making use of the ClausiusClapeyron relation (Chap, X)
■Bp
(3)
where  the specific volumes of the 'liquid and solid
respectively, and^~= the ratio of the change of pressure to the change
ol the freezing point. In this way L can be easily found.
10* Empirical Relationships.— It was observed by Richards that
if ML denotes the latent heat multiplied by die a torn it weight
melting point, ML/ T m is approximately constant for all
substances and its value lies between 2 and 3. This generalisation
is known as Troulon's Rule* But the relation is only approximately
true, and the value of the constant seems to depend upon the nature
of the crystalline form in which the substance solidifies. Table 2*
i how far this generalisation holds.
Table 2.— Illustration of Trouton's Rule.
Melting
ML
Substance
Atomic latent
point
T m '
Crystal system
heat, in cab
r m
Na
030
571
17
)
K
570
m
1*7
* Bodycentred
Rh
520
512
17
f cube
Cs
590
300
17
J
Cu
2750
135ft
203
j
Ag
Au
2630
3100
1234
1337
2 IS
282
I Facecentred
cube
Pb
1170
'.■■00
J W
Al
2500
930
270
J
Mg
1130
927
122
}
Zn
Cd
1800
1500
cm
594
260
253
/ Hexagonal.
560
231
: 240
[J
II, Effect of Pressure on Melting Point. Regelation— As already
mentioned the melting point of a substance is not quite fixed; it
I binges when the external pressure is varied. Equation (3) giving
»ani!;e in melting point due to pressure has been deduced in
Chap. X from thermodynamic considerations. This expression clearly
■ that the melting point of substances which expand on solidifica
II; n i. lowered by increase of pressure while the converse is the case
other class of substances,
.argcly taken from Hand buck der Experlmentoiphysik, Vol. 8, Part i,
CHANGE OF STATE
CHAP
Ice belc the first category, it is rfiis of ice which
accounts for the wellknown phenomenon of Regelation, i.r., the
melting o£ ice under pressure and its resolidification when the pressure
is released. This property enables us to explain the elegant expert
.i of Tyndall  in which a piece of wire loaded at either end with
weights anil placed on a block of ice finds its way through the latter
though the latter remains intact. The wellknown phenomenon of
glacier motion is partly due to the same cause. Snow goes on
positing on a mountain and when the mass attains sufficient height
the ice at the bottom melts under pressure and begins to flow, bur
as soon as the pressure is released it resolidifies. The block of ice
thus continuously shifts down the slope and we have the phenomenon
• i glacier motion,
12. Fusion of Alloys. — Alloys, except those having composition
in the neighbourhood of that of the eutecti* alloy, do not have a
j. finite melting point, Consider for example an alloy of lead and i in.
The melting of lead is S27°C and that of tin is 232°G, the
eutectic alloy having the composition 63^ Sn and &7% Pb. If an
alloy of a ana 10% Pb is cooled from the molten state it first
becomes pasty at about 210°C when solidiitcation commences and tin
begins to separate out, and this continues till a temperature of 1S$°C
is reached and the remaining liquid mass which has the composition
08% Sn and 87% l'h solidifies completely. Thus the addition of a
little lead to tin or a little tin to lead has the effect of lowering the
inciting point of the pure substance just as the addition of a little
melting point of ice.' In fan thp behaviour of the
alloy is just like that of me salt solution depicted in Fig. 1, Chap. VI.
i with a molten alloy rich in lead i.e., 80% Pb
the mass first becomes past) at about 275 °C when lead
parate out and finally the whole mass solidifies at, 183 C.
s whenever the alloy is very rich in one component, the molten
[11 first become pasty on cooling, the paste consisting of crystals
held in the liquid, and this will be indicated by a halt in the
cooling curve. On further cooling a second halt is reached when the
entire mass solidities at 183 4 C< The alloy corresponding to the corn
led the eutectic alloy and this tem
perature of ]iS;l' c C is called the eutectic temperature. Tf we start with
the alloy of this composition it will solidify or melt at a definite
ii nature — the eutectic temperature. Similarly if we melt an
alloy of composition other than the eutectic, it will first become pasty
and then at a higher temperature melt completely. The alloys of
Other metals in general behave similarly.
Alloys are of considerable practical importance. Thus ordinary
soft solder or tinman's solder is an alloy of lead and tin having about
60% Sn i.& a eutectic mixture of lead and tin. It has a melting
•John Tyndall (1820—1893), Born in Ireland, ed at Marburg and
in Berlin. From IS53 onwards he was Professor of Physics at the Royal Insti
tution in London, He was wellJcnown as a brilliant experimenter and was
i I of ::i::ni
VAPOR1SATION
i i than Lhat of tin or lead, and has a sharp and
nitc " i .n ure of solidification. An alloy of tin and lead (tin,
, i i, at 1J)4 6 C; Rose's fusible metal (tin, 1; lead, 1;
at 94°C; Wood's fusible metal (tin, 1; lead, 2:
I ; ii LLt ii. 4) melts at G0'5"C though the melting points
. and tin are 269, 327, 232°C respectively,
13. Supercooling— It has been stated above that when a liquid
i: lilies at a definite temperature (its freezing point).
wever, if slowly cooled in a perfectly clean vessel,
In down to a temperature much below the normal frcez
;iit solidifying. This is known as the phenomenon of
or surfusion. Water can in this way be cooled down to
still lower with a little care. Dufour suspended a minute
ma mixture of chloroform and of sweet almonds which
• i!i i gravity equal to that of tile water drop and managed
latter to — 20°C without solidification while a drop of
mplitlialene could be supercooled to 40 °G (normal melting point
«l C).
soling is, however, essentially an unstable phenomenon.
I I I'oduction of the smallest quantity of the solid in which the
I pud would freeze at once starts the solidification. Mechanical dis
h as shaking the tube., stirring or rubbing the sides with
■ i.l, or addition of some other solid is often sufficient to start
Lion. Tf solidification has once started it will continue with
ii ii nl heat till the normal freezing point is reached. After
ii Hi'!' solidification will take place only when heat is lost by
i libation, etc. Absence of air favours supercooling probably because
[usi particles contained in it are then absent,
VAPORISATION
14. Evaporation, Boiling and Superheating.— A substance can pass
the liquid state to the vapour state in two ways, viz (i) evapo
i ebullition or boiling. In the former the formation
es place slowly at the surface at, all temperatures, while
In the l.ni.'.'i the vapour is Formed in all parts of the liquid a! a
• i ii tant temperature and escapes in the form of bubbles producing a
Initially these vapour bubbles arc formed around small
nibbles of ah clinging to the bottom and sides of the vessel which
facilitate the process of boiling. If. however, the liquid is carefully
ii dissolved air and then heated in a clean vessel, its tem
i an be raised several decrees above 100°C without its begin
ning to boil, but when boiling starts due to disturbances of any kind,,
i ivith explosive violence, usually called bumping, and the
temperature falls to 100°C, This superheating is the cause of burnp
i prevented by the addition of porous objects.
15. Saturated and Unsaturated Vapours. — If the vapour escaping
1 a liquid either by evaporation or by boiling is collected in a
it will form an unsaturated vapour. In the case of evapora
in'ii in a closed space as in §2, the space will be filled with un
112
CHANCE OI STATE
CHAP.
saturated vapour if all the liquid has evaporated, but if the vapour
remains in contact with some liquid, it will he saturated. Unsaturated
vapour approximately obeys Boyle's law but for saturated vapour the
pressure depends only on' the temperature and not on the volume,
When saturated vapour is heated it becomes unsaturated or super
heated. When saturated vapour is cooled without condensation
occurring it becomes supersaturated.
We shall now describe some methods for determining the vapour
pressure. The range of pressure to be measured varies from 1G 4 mm.
to 400 atmospheres. In certain eases, pressures as low at I0* mm.
have to be measured. It is clear that such wide range of values
requires various kinds of apparatus.
16. Vapour Pressure of Water. — The first accurate determination
of iTnti pressure of saturated vapour was made by Dal ton. A similar
but improved apparatus was later employed by Regnault lor finding
the vapour pressure at temperatures
lying between 0° and 50°C. Ueg
ria nit's experiments were performed
with the greatest care and extend
over a wine range of temperatures.
His apparatus for the range 0° 50 c
hown in Fig. 3. Two barometer
tubes A and B were arranged side
side, fed from the same cistern of
M, The space a above the
mercury level in the tube B is
vacuum, while water is gradually
introduced at the bottom of A till
il rises through the mercury column
and evaporates on reaching h. More
water is introduced till a small layer
remains floating over the mercury
surface in A. A constant tempera
ture bath DD furnished with stirrer
(not shown) , and a thermometer
surrounds a, b f as well as some
length of the mercury column. The
dilTerence in the heights of the two
mercury columns, which ;<• _ observed
through a glass window with a catheto meter, gives the saturated
vapour pressure of water at the temperature of the hath. Correction
must he made for the weight of water in A, for effects of capillarity,
refraction, etc.
For re trip: ratines below C C "Rcgnault modified his apparatus to
ol Gav l.ussac. The top of the tube A was bent round and
• a spherical bulb which contained water or ire and "was
surrounded by a suitable bath. For temperatures not much above
50 C 'C the apparatus already described (Fig. §) could bt: used when a
Fig. 3.— Regnauli •
— "
;al methods
113
few
C
i;
loijfl ith would necessary, but Regnault preferred the boiling
tec. 18) .
17. General Methods.— The methods used for measuring saturat
i sure can be broadly divided into two classes •
in which the temperature is kept fixed and the
is determined either manomelxically or by measuring tire
1 s iturated vapour. This is called the direct or static method.
: Metliods in which the pressure is given and the temperature
ii It die liquid begins to boil is determined. This is the dynamic
18. Statical Methods. — Regnault's method is illustrative of class
i 1 ) The same method can be adopted For finding the vapour pressure
ol any liquid provided it ot react with mercury and the vapour
pressure is neither too high nor too low. A number of investigate
employed this method. Their apparatuses differ only in un
iitiai details. A general scheme of apparatus utilising this
i md is shown in Fig. 4. A is a small glass sphere, a
 i lacity, to which is attached a glass tube
incl another glass tube D with a smaller bore.
I In; is connected to a bigger globe G and a
ii' n ury manometer M about 90 cm. long. The
whole apparatus is first evacuated through the
itop cock Sj. and then the latter is closed.' Next
the gas under investigation is introduced
the stopcock S 2 and condensed in A by
cooling the latter; finally S 2 is also
The sphere A is then surrounded by
i] in me baths and the vapour pressure col
liding to the temperature of the bath is
a ted by the manometer M. B is a baro
to indicate the atmospheric pressure. The
apparatus is convenient for measuring vapour
tires from a few cm. to the atmospheric
For higher pressures a compressed air
ii 'meter may be employed when the whole
as to be made of steel,
\ similar apparatus was employed by Hen
I Stock for finding the vapour press >
ol a number of gases between f10 and
On the same principle Siemens has
I I vapour pressure thermometry at low
1 ' For high pressures we may
ii the classical experiments of Cailletet
"il' n (p. 101) with water, Andrews'
Apparatus (p. 88) may also be used, lluiborn
ind 13a ami determined the vapour pressure
"' watei ahovi 200 a C by the statical method,
[he methi also been employed by Smith and Menzies though
i] i is is ranch different.
.A
Fig:. 4.— Determination
of vapour pressure by
statical method.
1
114
CHANGE OF STATIC
CHAP.
Ill the experiment mentioned above the pressure was measured
manorjaetrically, but Uie pressure may also be found by determining
the density of saturated vapour, for assuming that the perfect, gas
equation holds, we have
pST
P^'
M
<*)
Thus knowing M, p we can calculate p. A very simple apparatus
based on this method was employed by Isnaidi and Gans. A portion
of the vapour was isolated and its density determined.
19, Dynamical or Boiling Point Method,— This method is based
on the fact that when a liquid boils, its vapour pressure equals the
external pressure on the surface of the liquid, A definite exter
nal pressure is applied on the liquid surface by means of a pump
and then the liquid is heated. The liquid will 'boil at the tempi
ture at which its vapour pressure equals tile external pressure,
that, set up by the pump. Hence the vapour pressure corresponding
to the temperature of ebullition is the pressure exerted by the pump
and can be read on a manometer.
Fig. S» — Iiegnault's Vapour Pressure Apparatus (Dynamical method.)
Regnault employed this method for finding the vapour pressure
of water betwe 50°C and 200 D C His apparatus is indicated in
Fig. ft. The copper boiler A is partly rilled with the experimental
liquid and contains four thermometers be immersed to different depths,
inside the vapour and the liquid. The upper part of the boiler is
BOILING POINT METHOD
115
cted by means of H to a pump, tfie pressure being indicated bv
n i cury manometer NM. The reservoir is kept immersed in a
bath VI and serves to transmit the pressure from the boiler to
' manometer as well a, to smooth the fluctuations in the pressure
til ined by the pump. The vapour of the liquid condenses in C
uul returns to the boiler, thus the same quantity of liquid is used
over and over again. First, a definite pressure is established by the
pump and the boiler heated. In a short Lime the readings indicated
bj i lie thermometers be become steady. The manometer indicates
trie vapour pressure corresponding to this temperature.
The apparatus can be adapted [or all pressures. For high pressure
■.'• ihe parts must be made of copper and the pump must be a force
SL,^^ is capable of great accuracy/ By this method
>rn and Henmng have very accurately determined the vapour
sure of water between 50 and 200°G ft to 16 atm.) . Ramsay aid
ug app^ed this method to the measurement of very small vapour
■■urts. J heir apparatus (Fig.
6) consists of a wide glass tube
i to which is connected a reser
• R, the latter being connect
to an air pump and a mano
meter M. Definite pressure is
'.<i up by the pump and the ex
ttal liquid, stored in F,
llowed to drop on the cotton
• 1 surrounding the bulb of
the thermometer. The tube T
kept surrounded bv a suffi
iy hot bath so that the ex
perimental liquid aL once evaporates inside T and the thermometer
soon reaches a steady temperature, the true boil.
mg point ot the liquid under that pressure.
Smith and Memies have devised an inireni.
ous modification of the boiling point method
Ineir apparatus is indicated in Fig, 7. The sub
stance under investigation is kept in the sphere
A close to the bulb or the thermometer T, both
"ttng immersed in some liquid contained in the
^tube 15. . The testtube is dosed airtight
and communicates with a pump and manometer
t shown) . It is further surrounded by a bath
whose temperature can be varied. A 'definite
pressure is first established by the pump and the
temperature of the bath gradually raised. When
the vapour pressure of the substance contained
m A becomes equal to the external pressure on
the surface of the liquid in the tube, any further
increase of temperature increases the pressure of
tire vapour m A which consequently bubble*
w
Fig. 6.— Ramsav and
Young's apparatus.
JJc=*
'''" i and
apparatus.

116
CHANGE OF S'
[ CHAP,
\ .
LATENT HEAT OF VAPORIZATION
117
through the liquid in the tube. When this just happens the press un
recoX by the manometer ^ the vapour pressure corresponding
to the temperature indicated by the thermometer.
The boilingpoint method has also been applied to metals by
£e .."cadmium was heated electrically in a quartz or por,
Jeltin tube and the temperature of the vapour was measured by a
thermocouple.
20. Discussion of ResultExperiments show that the saturated
vapour pressure of every substance increases as the temperature n
raised. Hence the vapour pressure f> must be a function oi the tem
perature viz ft — f IT) where / (T) must be of such form that its % aluc
E522 Sh "the' Utperature T, Various empirical relation*™
between ft and T have been proposed from time bo time. They hold
for limited ranges and are by no means quite exact, and universal.
In 1820 Young proposed a very simple formula
log p = A + y>
and
are constants.
where T is the absolute temperature ana A,
KirchhoR in l I Rankine in 1866 proposed quite independent^
the form li la
B
\tygp =A+ r fClogT.
m %
This formula agrees with experimental results very closely and can
also i„. il considerations. It is shown in
Chai that
dp_
a)
e fs h the vapour pressure. L tbe latent heat of vaporization, T
loint and v* v t , the specific volumes of the substance
in the liquid and vapour states. Neglecting v x in comparison with v s
v 2 by R T/Mp from the gas laws we get
.  • (8)
lo g p= R \ Y ,+u
Tf.
Assuming £ constant, equation (8) yields Young's formula (5) .
•::v£ v. , we assume L to vary linearly with temperature, i.e.,
L = L n — i*Tj,
equation {9) yields us formula (6) . In order to get the exact, value
r l' pressure corresponding to any temperature we must use an accurate
expression giving the v due of L as a function of T,
The above holds for the saturated vapour pressure of a pure
liquid; when mixtures of two liquids are investigated they yield bi
ting results, (1) Tn case of liquids which do not at all mix the
:. ii pressure of the mixture is equal to the sum of the vapour pres
sure of the constituents, e,g*, water and benzene or water and carbon
(2) Iii case the liquids are partially miscible the vapour
in e is always less than the sum of the vapour pressures of the con
s and may even be less than that of one of them, e,g., water
.iii.l ei tier, or water and isobutyl alcohol. In this case each constituent
Little of the other and the vapour pressure of the mixture
aains constant over a large range of composition of the mixture but
falls off for very dilute solutions finally attaining the limiting value for
the pure constituent. Thus if we distil a weak solution of isobutyl
alcohol, we find that the solution in the still becomes weaker and
I finally there is only pure water in the still and pure alcohol
in i he condenser! For a weak solution of water in alcohol the reverse
ase. (3) When the two licjuids are wholly miscible, e.g., water
and methyl alcohol, the pressure is intermediate between those of the
arate constituents. In this case whatever composition we start with,
alcohol always passes to the condenser leaving pure water in the still.
21. Vapour Pressure over Curved Surfaces. — In the foregoing we
have considered the vapour pressure over a flat surface. The vapour
pressure over a curved surface is different on account of surface ten
i. Evaporation from a spherical drop produces a decrease in
area and hence also in the surface energy due to surface tension
and therefore, it will proceed further than in the case of a flat surface,
i.e., the vapour pressure over a convex surface will be greater than
that over a fiat surface. Detailed considerations yield the result
taA.^1 (9)
re fa, p denote the vapour pressure over flat and curved surfaces
respectively, S the surface tension* r the radius of curvature of the
rface (considered positive for concave and negative for convex)
and p tjie density of the liquid. These considerations have important
nsequenr.es in the precipitation of rain and in the phenomenon of
[]i v. Equation (9) shows that if r is small and n* Le. 3
rface is convex, p may become very large. Hence, if suitable
nuclei for condensation are absent a high degree of supersaturation
■ he attained and inspite of it, no drops will be formed,
LATENT HEAT OF VAPORIZATION
22. Tn the measurement of latent heat of vaporization we have
i" remember two points; first, that k?\ absolute value is relatively
'y, secondly, that the latent heat is absorbed or evolved in an
isothermal change of state. The consequence is that its experimental
deterj lination is very little affected by the usual sources of error which
isent in all ealorimetxic measurements. The methods can be
grouped under three broad headings:
l) Condensation Methods, — Those in which the amount of
I it ".ii evolved when a certain amount of vapour condenses h measured,
(B) Evaporation Methods. — Those in which the amount of heat
requ i vaporize a given mass of the liquid is measured directly.
"1
118
CHANGE OF STATE
[ CHAT*.
v
EVAPORATION ME1 I
119
The heat is generally added in the form of electrical energy and can
1 easily determined.
(G) Indirect Methods. — Those in which the la Lent heat is cal
culated with the help of some thermo
dynamical relationship such as Clan
GJapeyron relation (equation 7) . From
the vapour pressure curve the quantity
dp/dT is determined and hence L
evaluated from (7). We shall now con
sider the first two methods in greater
detail
A. Condensation Methods
23, Berthelofs Apparatus, — Reg
nauit's experiments are rather of histo
rical interest We describe below the
apparatus ol' Berth Hot (Fig 8) , In this
apparatus which is wholly of glass the
liquid is kept in the vessel D and heated
by the ring burner B, The rest of the
apparatus is protected by an insula
mantle M, The evaporated gas passes
through the tube T into the" spnal S,
placed within the calorimeter C. The
iral S is fitted to T by a conical
ground pio an be easily remo 1
The vapour condenses within the its latent heat
lorimeter which can be easily measured by
observing the rise of temperature on die thermometer placed inside a
jacket. The amount of water condensed is obtained by welch
ing the spiral S before and after the experiment The heat, measured
the heat of vaporization plus the heat given by the con
I liquid in cooling from the boiling point to die final tempera
ture of the calorimeter. The open end or S is connected to a pump
to regulate the pressure under which the boiling takes place.
Errors are likely to arise owing (1) to superheating of the liquid,
(2) to minute drops of wai ing carried over by the vapour. As
the: use of a ring burner causes the heating to be sometimes irregular,
Kahlenberg replaced the ring burner by a metallic spiral placed inside
the liquid and heated electrically.
24, Awbery and Griffiths used a slightly .modified apparatus in
which the usual calorimeter was replaced by a continuous flow calori
meter. The apparatus is shown m Fig. 9. The boiling chamber is
heated electrically by an inner coil. The vapour passes down die
vertical tube which is surrounded by a jacket of water through which
a stream of water flows at a constant rate. The temperatures of the
inflowing and outflowing water are determined by two thermocouples.
There is a third thermocouple at the mouth of the vertical tube which
L— Berthelot's .
• LUS.
\ Ijjpti htasins «fl
iij?3ic&Gntei>i}
gives the temperature of the condensed liquid as it leaves the appnra
In this experiment the
. i pour must be produced at
a stead 1 .' rate and this is
achieved by the use of elec
trical heating. The latent
at is obtained from the
i" mula
M8=m[Lt*(t s —t l )l (10)
where 9 is the excess of
temperature of the outflow
water over the inflow water,
M is the quantity of water
flowing per unit time, m is
the rate at which the liquid
is being distilled, t 2 is the
boiling point of the liquid,
,*, the temperature of the Ii
td as it leaves the appara
tus.
B. Evaporation Methods
25. This method wa i
employed by Dieter id. who
measured the heat required
to evaporate a given mass of
water with the help of a
fiunsen icecalorimeter. The
water was contained in the
tube A "of the ice calorimeter (Fig. 3, p. 34) and the heat was measured
by finding the mass of mercury expelled. Griffiths found the electrical
energy required to vaporize a given mass of water. We shall describe
the apparatus used by Henning For precision measurement of the heat
of vaporisation between 30 p and 100°C.
26. Henning's Experiments. — The apparatus employed by Hen
ning is indicated in Fig. 10. C is a copper Vessel, one litre in capacity,
in which the liquid is allowed to evaporate. This is surrounded by
an oilbath A maintained at a constant temperature. The heating
! place through the spiral D of constantan wire wound on a
quadrilateral mica frame. E is a platinum resistance thermometer.
Tfu vapour which is evolved passes through the German silver tube
H downwards through the German silver tube KK to the vessel P in
which it is condensed and weighed. The end of H is bent down
wards so that no liquid drops can be carried. The vapour is first
! Is P, and when the conditions become steady
h i opcock R h turned so that steam is led to the other vessel P.
tfter a sufficient quantity of steam has been led to P the stopcock
turned to the other side.
Fig. 9.— Awbery and Griffiths* Latent
Heat Apparatus.
120
CHAKG£ OF STAT:
[ CHAP
The quantity of vapour deposited in the second vessel is now
Lou id b) weighing and the quantity
of heat ' supplied " is obtained from
observations of the electrical measur
ing apparatus. For determining; the
heat of evaporation at lower pres
sure, P is connected to a large vessel
i i about 5 litres capacity which is
maintained by means of a water
pump at the required pressure,
A similar apparatus was em
ployed by Henniiig Tor finding the
heat of vaporization of water up u>
I80°C when the pressure reaches
about 10 atmospheres. Fogler and
Rodebush used this method for deter
mining the latent heat of evaporation
of mercury up to 200
For determining the Intent heal
of evaporation of substances like
nitrogen, hydrogen, helium, etc. r
which become liquid at very low
temperatures, the above principle
has been utilized by #Dana and
• r . . t_r * Ormcs, Simon and Lance and
_ieat °
others.
DISCUSSION OF LATENT HE J DATA
27. Variation of the Latent Heat with Temperature. — Experimenl
; : ::j latent heat diminishes as the temperature at which
boiling takes place is raised. This was no v&a by early
ors who proposed various empirical foruuilae. Of these
Thiesen's formula appears to
have been most satisfactory
and states
L^L X (*,!)*  (11)
where t t is the critical tempe
rature and Lj a constant winch
tfces the value of L at
t = t &  1, This is of course
based on the assumption thai
latent heat vanishes at the
critical temperature, Henning
showed that between 30° and
100C the latent heat of vapo
rization of water is given by
the formula
L= 538.86+05994(1000. (12)
Fiff, 11,— Variation of Latent Heat
of CO j with temperature.
trouton's rule
121
._itrgator.
shows dearly that the latent heat vanishes at the critical tempera.
e of CO, which is about SPG This result is quite, universal.
e exact variation of latent heat with temperature for all substances
iven bv the thermodynamic formula
where c e) c s denote Che specific heat of the substance in the gaseous
and liquid states respectively and v it v t , the respective specific
volumes,
28. Troirton's Rule*— As in the case of latent hear of fusion,
we have here also an important generalization known as Trouton's
Rule which states that the ratio of the molar latent heat arm
the boiling point is a constant for most substances; or
symbolically,
&L= constant, .... (14)
where M is the molecular weight and Tt, the boiling point. The
ue of the constant is about 21. The law does not hold for asso
rted vapours. Table 3 shows that tire law holds approximately for
mc;;: substances,
Table 3 — Illustration of Trouton's Ride.
Substance
Gra mmol ecul ar
latent heat in
en lories
Ml
ii Pin
point
T, v
429
204
773
961
1881
8595
8090
319
353
:■:■
6800
858
942
1155
1180
1887
v , r ML
Value O: ,.
1 b
experimentally
Helium
Hvdrogen
Nitrogen
Oxygen
ochlonc acid
Chlorine
: anc
Carbon disulphide
Benzene
Aniline
Mercury
slum
"Rubidium
Sodium
Zir.c
Lead
22
219
1310
1030
3890
4600
6100
6490
7350
10000
14200
15600
18700
29300
27730
4f)000
51
10
173
181
207
.
1975
208
210
226
182
199
202
235
244
T
122
CHANGE OF STATE
[ CHAP,
V.
DENSITY OF SATURATED VAPOUR
123
It will he seen, however, that ML/T is really not quite constant
for all substances, A simple theoretical discussion shows that it can
not be so., for the boiling point under atmospheric pressure is purely
an artificial point and has no physical significance. It varies enor
mously with the external pressure, while the latent heat, varies neither
in the same direction nor to the same extent. In the case of water
the Trouton quotient at a few temperatures is given in Table ■:.
Table 4. — Trouton quotient for water at nt tempera? urex.
Pressure
46 mm.
760
latent heat Trouton quotient
6065 cal.
;
40
259
16
From the variation in L and the boiling point it is evident that
the Trouton quotient will go on decreasing as the temperature
becoming zero at the critical temperature where the latent heal
rushes. Thus the quotient can have any value from. to 40 and
it appears to be merely an accident dim • t substances boiling
undi the value is about 20, #
DETERMINATION OF VAPOUR DENSITY
29, sity we mean the spec the vapour
lir or hv ity. The vapour density can be ea
known weight of vapour
ai a certain pressure and temperature. Thus if to grams of die
v c.c. at the pressure p and temperature j", and p rt
it of .. of the standard substance (air) at 27i' IJ and
760 mm, then the vapour density is given by
w T_ 760
/V 273 'p W
The density of the unsaturated vapour can be easily and a ecu
rately found by any <.uiq of the standards methods* viz. ? of Victor
Meyer, Dumas, lio'fmann and others. But before I860 there was
no method for directly determining the density of saturated vapours.
The methods adopted were all indirect in which the density of the
unsaturated vapour was first determined, and assuming the perfect
gas laws to hold up to the saturate:..] state, the density of' the saturated
vapour was determined. This assumption is, however, not quite jus
tifiable, hence these methods can never give accurately the density of
iturated vapour. Still, however, they are frequently' used especially
*Fufl details of these methods will be found in any textbook on Physical
Uxemistry.
dial of Victor Meyer, which is of considerable practical importance
is consequently described below.
The apparatus (Fig, 12) consists of a
cylindrical bulb B with a long narrow stem,
ttear the top of which there is a sidetube.
The lower part of the tube is surrounded
i suitable temperature bath which is
.' t epl nished by a suitable liquid boil
ing at some pressure in A. Air inside the
tube gets heated and is expelled at the
top; after a time, however, a steady state
is attained when no more air escapes. The
substance whose vapour density is to be
determined is enclosed in a thinwalled
stoppered bottle and placed inside the
tube. By manipulating s it is allowed to
: gently in 15. The bottle breaks, the
liquid vaporizes and thereby displaces
an equal volume of air which escapes at
the sidetube and is collected in the tube L
Knowing the mass of this air, rhe density
of fl r of the substance is obtained
the mass of the liquid taken
by the mass of displaced air.
rnst has modified the apparatus and
could thereby measure the vapour density
of KC1 and NaCl up to 2000°C while Wartenberg found the vapour
density of several metals up to 200f.PC, Foe a
description see Arndt Pkyxifialiskchernische Tech'
mk. Chap. IX.
30. Accurate determination of the Density of
Saturated Vapour* — In 1860 Fairbairn and Tate
sed an apparatus by means of which they mea
sured the density of saturated vapour directly.
Fig. IB explains the principle of their apparatus, A
is a spherical glass bulb whose narrow stem dips
into mercury contained in the outer wider glass tube.
I r communicates with the metal reservoir B.
Both A and B contain some water a hove the mercury
levels, the latter containing a larger quantity than
the former. All air is expelled from the apparatus
and then both the vessels are surrounded by a bath
whose temperature is gradually raised. The levels
of the mercury in the two vessels remain con.
that in A being always higher than in B due to the
excess of water in B. This is so as long as there i%
any liquid water in A. But as soon as the liquid
in A disappears, the level of mercury in A suddenly rises. This is
because the saturated vapour pressure increases much faster than the
Fjjt. 12.— Victor Mej
Vapour Density Apparatus
Fig, 1&— Fair
bairn and Tate's
apparatus
124
CHANGE OF STATE
CHAP.
I
pressure of unsaturated vapour obeying Boyle's law. The tempera
ture at which this sudden rise of mercury column in A appears is
noted. At this temperature the vessel A becomes filled with saturated
or whose pressure may be found by means of a gauge connected
to B. Knowing the mass of water in A and the volume of the
enclosed space, ihe density of the vapour at the particular tempera
ture and pressure can be calculated.
Somewhat later Perot attempted to isolate a portion of the
saturated vapour and to weigh it. This could be very conveniently
done by isolating the vapour by means of a stopcock and getting
it absorbed in dry calcium chloride and weighing the latter. K. Qnnes
employed another simple method. In a graduated vacuous tube
different masses of the liquid are introduced and the volumes of the
mr and the liquid observed. Thus if m and m' grams of the
substaj introduced and the volumes occupied by the gas and
the liquid are Vi and v» in the first case, and v\', v? in the .second
case, and p g , p t represent the densities of the gas and die liquid, we
have
whence
•—'•■; i ,
«'=*»!>* +P»'W,
"r.'^ r 'i. "t
(16)
Book Recommended.
azebrook, of Applied Physics* Vol. 1 1 article
CHAPTER VI
PRODUCTION OF LOW TEMPERATURES
I7e low^nP^wVc obtainable. The study of the laws o per
SrC can, at least theoretically, proceed on the «n
£5fe*te SJ degree, below the ^eUing point of rcc ; a, ;1 £.
i?.he lowest temperature conceivable. This is taken as the zeio
of tne aSte^mperaUire scale. In this chapter we shall dtscu^
die principles and contrivances by which die region from C to
absolute zero can be reached.
PRINCIPLES USED IN REFRIGERATION
2. For reaching low temperatures we have to utilize proc, ;
bv which a bodv can be deprived of its total heat content. Ihe tol
lowing methods' may be employed to achieve this end :—
(i) Bv adding a salt to ice.
m Bv boiling a liquid under reduced pressure.
hit) By the adiabaiic expansion of a gas doing external work.
(iv) By utilizing the cooling due to JouleThomson effect.
(v) By utilizing the cooling due to Peltier effect,
(wi) By utilizing the heat of adsorption.
(vit) By the process of adiabatic demagnetisation,
A general theory of refrigeration will be given later in Chap, IX;
here we shall simply discuss the principles and contrivances for
utilizing them.
(i) Adding a Salt to Ice
3 Low temperatures may be attained by adding a salt to
Ice This is the same process which was employed by Fahrenheit.
The cause of this lowering of temperature is easily imdei
Pieces of ice have generally some water adhering to them, and
if salt be added to this ice, it is dissolved by the water and
more ice melts. The necessary heat for this process, viz., the neat
of solution and the latent heat required to melt the ice, n extra.
from the mixture itself whose temperature consequently falls own.
126
PRODUCTION OF LOW TEMPERATURES
[ CHAP.
This is the principle of pvezing mixtures. This process, ho
owever.
cannot go on indefinitely. Fig. 1
shows the freezing curves "ob
tained with ammonium chloride,
the ordinate representing tem
perature in °G and abscissa V
concentration of the salt. When
the salt is added to ice the tem
perature of the mixture chan
as represented by the Hine AB
1 the eutectic temperature of
~]5<8°C is reached. Tempera
tires lower than this cannot he
obtained in this way for when
more salt is added it no long
into solution. The curve
AB represents equilibrium bet
I'tff. L— iNtua aad water ■ ■ ., ; e n solution and ice while CB
™n i i .. ^ „ , represents equilibrium between
salt and solution and B denotes
e eutectic mixture with a Table 1. — Freezing Mixtures.
ed composition and fixed tem
perature.
In Table 1 the composition
of the eutectic mixture and the
corresponding eutectic or c
tempera
wnmoner salts, i
rally hydrated salts are employed
! in that case the oorre!
ing quantity hydrated salt
Id be obtained by calculation,
i raperatures repre
the lowest temperature that
is possible to attain with that
Freezing mixture,
(if) Boiuwg 4 Liquid Under Reduce Pressure
4. Lot ntme may also he attained by allowing a lie
boil under reduced pressure. When a liquid evaporates it requires
U for conversion from the liquid to the gaseous state (latent heat
of vaporisation) Thus one gram of water at 100*C requh
calories for complete evaporation. If such liquid be forced by so]
conmvancc to evaporate rapidly and if the liquid be isolated." rh."
ig may be produced.
The oldest contrivance for utilising this process is the crvo
Wicated in Kg. 2. The bulb B contains water or some
volattle liquid and the rest of the space is filled with the lap u?
1 Aulr.
En:.
Salt
! saJtperlOO
tempera ,
grams of the
lure.
rture.
; o 4
197
^f
ZnS0 4
27^
.65
KC1
197
 T T • 1
NH 4 CI
1 5  S
iNO s
412
174
o 8
37
185
NaCl
 21 2
: :i a
21*6
336
CaCL
29 B
 V.
KOH
81*5
<o5
VAPOUR COMPRESSION MACHINE
127
of the same liquid. When A is immersed in ice the vapour in it con
denses and the pressure of the vapour in B becomes so much lowered
that the liquid 'In B boils. The latent
heat necessary for this purpose is ex
tracted from the rest of the liquid which
consequently free.
Xowadays this principle is em
ployed in a huge number of refrigerat
ing machines both for industrial and
domestic work. Water, however, is not
a suitable liquid to use for though it has
a large latent heat of evaporation, the
vapour pressure at low temperatures Fig. 2.— The Cryophorus,
is small The liquids commonly employed are ammonia, sulphur
dioxide, etc. Two types of such machines arc in use : — (1) Vapour
compression machines, (2) Vapour absorption machines. The vapour
compression machines are more efficient, particularly for large plants,
and require less initial cost; consequently, their use is 'more common
than that of the other. The only essential difference between these
two types of machines consists in the manner of compressing the
low pressure vapour. In the former a motor compressor is used while
in the latter a dilute aqueous solution at ordinary temperatures is
employed to dissolve the low pressure vapour and the coneentral
solution is heated in a generator to expel the gas at high pressure.
We shall now describe these machines in greater detail.
5. Vapour Compression Machine.— Fig. 3 shows the essential parts
oE a vapour compression machine. There are three principal parts
Water ou
P=1L Etmos js ,
CoMsto
apftse
p=2'5 atmoa.
p~j] Liquid ammonia
pffil High pressure ammonia
,ow tiresaure ammonia
V
Wig. 3. Essential parts of a Vapour Compression Macbin
the compressor P, the condenser C and the refrigerator or evapo
rator R. The cylinder of the compressor has two valves, S and D,
.28
PRODUCTION OF LOW TEMPERATURES
[ CHAP,
rhe former for the suction of the low pressure vapour from the eva
porator and the latter lor the discharge o£ the compressed vapour
to th< condenser. When the piston p moves upwards the pressure
in the cylinder falls below the pressure in the evaporator and
the l™ pressure vapour Is sucked in through S and the suction
pipe. " Dicing the downward stroke Lire vapour is compressed and
then delivered to the condenser C through the discharge vake D
and the discharge pipe. The condenser is cooled by cold
dilating in the outer chamber. On account oE the low ^temperature
and high pressure the vapour liquefies in C. This liquid passes
through the expansion valve or the regulating valve V which is simply
a throttling valve to reduce the pressure of the liquid from the high
pressure prevailing in the condenser to the low pressure m the eva
porator Due to die low pressure the liquid boils thereby extracting
its latent heat from the cold storage space surrounding the eva
porator. This space is consequently cooled. In some cases the
evaporator is surrounded by brine water kept, in circulation. The
brine water thus becomes cooled and is taken elsewhere for reFrige
rati i purposes. The low pressure vapour is sucked in by the com
pressor and the cycle or operations continues.
In the diagram anhydrous ammonia is supposed to be uti
efrigerant. The pressures and temperatures of ammonia In
of die apparatus are approximately as indicated in the
.
i : on machine is shown in Fig. 4
w hicl Iphur di 13 the refrigerant. Vapour
ively employed in icemaking.
t and other foodstuffs and for various other Indus
trial purp
6. Refrigerants.— Various liquids have been used as refrigerants,
important one ai imoniaj sulphur dioxide, ethyl chloride
methyl chloride. Of these ammonia is most commonly used in
refrigerating plants, while sulphur dioxide is employed in many
household , There are various criteria for .selecting a suitable
refrigerant : (1) The latent heat of the refrigerant should be large
so that the minimum amount of liquid may produce the desired
refrigerating effect. (2) The refrigerant must be a vapour at ordinary
temperatures and pressures but should be easily liquefied when com.
d and cooled. Generally a temperature of about 5°F (some
below icepoint) is required in the evaporator coils and about
(about room temperature) in the condenser coils. (3) The pres
1 of the vapour of the refrigerant in the evaporator coil must be
greater than the atmospheric pressure, so that atmos >' impurities
may not be sucked inside and later block the valves. With cooling
at room temperature surrounding the condenser the pressure
necessary to liquefy the gas in the condenser should not be too large
otherwise the compressor and the cylinder will have to he made very
stout and consequently costly and there will be much leak g
vapour into the atmosphere/ (4) The specific volume of the vapour
I
VI.]
REPlUOERANTS
129
of the i ••:: 1 l should not be large otherwise a very large compressor
will be necessary,
. 7 h  :: important pii  me common refrigerants are given
m rable 2. V, is table it k evident that ammonia is the
gerant. One pound of ammonia will produce
the same amount of refrigeration as 8.75 pounds of carbon dioxide
while the pressure hi the condenser in case of carbon dioxide is about
ater than in the case of ammonia. In the matter of
spin j! 1 volume, however* carbon dioxide possesses an advantage.
Sulphur dioxide requires a less stout compressor and condenser than
rua but for the same refrigerating effect the compressor ha
be made large.
Table 2. — Characteristics of refrigerants*
Ammonia
Sulphur
Carbntl
' dioxide
Methyl
irjde
Freoti
CCJiF.
1. Boiling point in
°F at a tm. pres
sure
28.0
14.0
108.4
10.6
21.7
2. Latent heat of
1 \ aporation at 5
in B.t.u. per
pound
565,0
175.fi
114.7
178.5
69.5
5. Refrigerating
effect in B.t.u
per lb.
474,
141.37
JUS/;!
148.7
5 LOT
L Vapour pressure
at 5°F in lb/in s .
34,27
11.81
331.4
20.89
26.51
5. Vapour pi esi
at 86°F in lb/in*
6. Specific volume
of vapour in
e vapor?
cm ft. per lb.
7. Horsepower for
a ifrigei Ei n ]
200
B.t.u. per min.
60,45 10B9.0
6.421
0,09
>3.2
95.53
4.529
1.06
107.9
].1S
. : :
* A number ants have been
180
PRODUCTION OF LOW TEMPERATURES
[CHAP.
This method mav be employed to obtain extremely low tem
peratures by using liquid hydrogen and helium. These liquids are
Sowed to boil under reduced pressure when temperature* lower than
their normal boiling points are reached,
7. Electric Refrigerator,— Fig. 4 shows a modern electric refri
. l . Y?  ' „t „:«..■ .nwlin™ iiitniYMiirailv on r. he vaUOUr
(Oil
Liatricl SO^
I High pressure S0 3
I Low pressure S0 3
Fig. 4— Frigidaire,
The refrigerator coil R contains liquid sulphur dioxide which extracts
he:!, from the surrounding space and evaporates and the low pressure
vapour collects at the top. This vapour communicates i wtti i the
suction pipe S and the crank case K to the motor switch W. When
eno has collected in die top of R it exerts a large pressure
which is transmitted through S and K thereby operating the switch
VI.
AMMONIA ABSORPTION MACHINE
131
'.,'.'. This starts die motor and the latter works the compressor P, as
a result of which, the low pressure vapour is sucked in through S to
the ciunk cs and compressed by the piston and delivered to the
condense] C. The condenser is cooled by a current, of air forced
across il b) the fan mounted on the motor M but in some cases the
cooling is bi ought about by the flywheel itself whose spokes are shaped
like the blades of a fan. The high pressure sulphur dioxide vapour
on his cooled liquefies and collects in the reservoir T. From
here on > l of the high pressure the liquid is Forced up through
the liquid pipe L and enters R through the needlevalve N. When
enough liquid, has collected in R the float valve V rises and closes
the valve N. Thus the machine works only when the gas
essure in R becomes large and liquid is transferred from the storage
ik T to R only when the quantity of liquid in R becomes less and
float valve has sunk so as to open the needle valve N.
8. Ammonia Absorption Machine. — 4j> already slated the ab
sorption machines differ from
the vapour compression ma
chines only in the manner
of converting the low pres
vapour into high pres
sure vapour. Ammonia is
the most suitable refrigerant
use in absorption ma
chines, and water is a very
suitable absorber. Water at
F absorbs about one thou
i id times its volume of
ammonia vapour but when
the aqua ammonia 'solution is
freely escapes from the solution.
The wolfing of >an ammonia absorption machine will be easily
understood from Fig.J5. The generator A contains a strong solution
> nonia gas in water and Is heated by a burner as shown in the
or by means of pipes carrying steam. Ammonia gas is expelled
from the solution and passes into the coils immersed in the con
denser 1J through which cold water continuouslv Hows. The; gas
idenscs there under its own pressure into liquid ammonia. The
liquid ammonia thus formed passes through a narrow regulating valve
to the spiral immersed in the refrigerator C, where on account oE
ili low  i me it evaporates. The valve is adjusted to maintain
desired difference of pressure on the two sicles. Through the
refrigerator flows a stream of brine water which becomes cooled by
the evaporation of ammonia. The cool brine colution may be taken
to .il i \ place for refrigerating purposes.
mmonia gas formed in the coils in C is absorbed by water
i diluta ammonia solution contained in the absorber D arid thus
the pressure is kept low. The solution in D hecomes concentrated
and is transferred by the pump P to the generator at the top. Thus
I" ... 5.— Ammonia Absorption Machine.
heated to 80° F ammonia vapour
132
PRODUCTION OF LOW TEMPSEEATUREfi
P
the supply of concentrated ammonia solution is kept up. Dilute
ammonia from tin bottom of the generator may be run to the ab
sorber and concentrated. Thus tlie cycle is repeated and the action
is quite continuous.
The difficulties of this machine are that It h a m\\ efficiency
and the pressures are widely different in the condenser and the
orator. The low efficiency h due to the circumstance that the
heat absorbed by the ammonia in the generator is much larger than
the heat absorbed by it in the evaporator coils. Further the
machine has a moving part in the pump needed to transfer die
liquid to the generator, and is costly. All these difficulties are
avoided in a clever invention by two Swedish engineers, von Platen
and Muuters, which is :<3 in the market under the name Elec
trolux Refrigerator, hi this Dalton's law of partial pi. is used
to make the tola tire in the condenser and the evaporator
equal, maintaining at the same time a difference in partial pressures
of ammonia in the two chambers ; this is accomplished by using an
inert gas like hydrogen at a pressure of 9 atmos., the partial pres
sure of ammonia being •:! atmos. in die evaporator and the absoi
and ammonia liquefies in the condenser at the pressure of 12 atmos.
Concentrated ammonia solution is forced up into the generator by
heat and not by a pump.
(iii\ ^diasatic Expansion osf Compressed Gases
9. Cooling produced by the sudden adiabatic Expansion of com
pressed gases. — ■ Idenly allowed to
considerably on account of the
k it docs in expanding, vide' § 23, Chap. II v
////?) 'v ' ■>'>: The cooling: may he so great
that the ga.s may eve [£y.
An example which is easily available in a big town is afforded
i irbon dioxide. If such a cylinder be sud
denly opened and a piece of cloth held before it, the issuing ■
deposited in the form of solid CO a (called dry ice commercially) .
This principle was utilised by Cailletetwho first liquefied oxv
in 1877, He compressed oxygen to a pressure or 300 atmosphere
j capillary tube cooled to 2SPC by liquid sulphur dioxide boil
rider reduced pressure and then suddenly released the pressure.
A mist, of liquid oxygen was formed which disappeared in a few
seconds. PicteL compressed oxygen to a pressure of 500 atm. and ec
it to about — 130°C by liquid carbon dioxide evaporating under re
re. Then he suddenly released the pressure. Oxygen En
the form of a white solid was thereby obtained. Tn 1884 Wrob
obtained a mist of hydrogen while in 1S93 Olszewski obtained liquid
tin ient quantities, by cooling 'compressed hydrogen with
liquid oxygen and then suddenly releasing the pressure. Simon in
1933 produced appj quantities of liquid helium by sudden adia
•jiiic expansion of the compressed gas which had been precooled by
solid hydrogen evaporating at reduced pressure.
VL]
[PACTION OF GASES
133
The process is, however, essentially discontinuous, hence for
commercial purposes it was almost discarded; but a novel wr.
utilizing the principle has been invented by Claude and Heylandt
foi I air (see sec. 22) »
W; '"escribed above two types of refrigerating machines.
There is a third type also which is sometimes employed. This may be
called the air compression machine because air is here used as the
[rigerant. In dlis air is first comprcsssed in a compressor, the heat
of co ion is then removed by passing the gas through coils kept
, cold circulating water. This cool compressed air then suffers
adiabatic expansion in the expansion cylinder and becomes consider
ably cooled. This cold air then traverses the cold storage space and
thu Mated and is again compressed. Thus the cycle continues.
This is the principle of the BellColeman refrigerator largely used
for the refrigeration of cold storage chambers in ships.
(h) COOLING DUE TO PELTIER EFFECT
10. It is well known that when an electric current (lows in a
circuit from bismuth to antimony through a junction, this junction is
ooled. Ellis is known as the Peltier effect and may be utilized in
tducing cooling. This cooling is rather small though semiconductor
thermojunctions have recently produced much more cooling, and have
been employed in some refrigerators.
(*■•) Cooling by JouleThomson Expansion
11. This method, is of considerable importance and will be con
sidered in detail later in this chapter.
(vi) Cooling Due to Desorftion
I2» Charcoal adsorbs a number of gases which are released on
lowering the pressure, and when these gases escape, a cooling results
a manner somewhat analogous to the case of evaporation of liquids.
is is called the "Desorption method" and was utilised by
i in. In an experiment charcoal adsorbed helium gas at 5 atm.
I0°K and subsequent de sorption to 0.1 mm. pressure lowered the
temperature to 4 P K which is sufficient to liquefy heliu: l.
LIQUEFACTION OF GASES
13. Liquefaction by application of pressure and low temperature. —
which are gaseo^ at ordinary temperatures may be
i the liquid state if'they are sufficiently cooled, and sirnul
 pressure be applied to the mass. When pressure
the molecules come closer together and if heat motion
sufficiently small, they may coalesce and form a liquid mass,
ll the liquid so cooled be allowed further to evaporate rapidly,
Lem nines may be obtained. The production, of
in' I leratures is thus intimately connected with the
. ■ i : hi of such gases which ordinarily show themselves
efracl
134
PRODUCTION OF LOW TEMPERATURES
[CHAP.
compression was
The earliest scientist to try this effect of combined cooling and
Faradav* who, as early as 1823, employed the
apparatus shown in Fig. o
for liquefying chlorn
One end of the bent glass
tube contains dr.: substance
from which chlorine is
liberated by heat while the
other end is immersed in a
freezing mixture. Gaseous
chlorine coltecLs in the
other end and finally lique
fies under its own pressure.
By applying this process
Faraday and others suc
ceeded in liquefying a large
number of gases but some
viz., oxygen, nitrogen, hy
drogen, carbon monoxide
and methane baffled al! at
tempts at liquefaction,
to S000 arm. were used.
Fig. &— Faraday* :itus iot Liqpefeclton
of Chlorine,
carro
in 1! abject
And i'
w
abov
to which it is .subjected.
though sometimes enormous pressures up
They were, therefore, termed menl gases.
Discovert of the Critical Point— Ths cause of these failures "be
ta Andrews' discovery of the critical temperature
has already been treated in Chapter IV.
arty showed the importance of the
Ti, ■ 1 1 clea rly that, for ev er y subs tance ,
Wchmudlyo'ccurs in the gaseous form, there exists a temperature
bove which 'it cannot be liqueSed however high mas' be the pressure
, w h subjected. Hence, in order to liquefy a gas by this
method it must be precooled below its critical temperature.
The determination of the critical point (p, 100) is not, how
ever, easy. The early workers did not in fact, wait for its deter
mination. They cooled the gases by ordinary methods as much as
they could, add then applied high pressures.
' 14. The principal methods of liquefying air and other gases are
the following:— (1) Pictets cascade method which utilises a series
of liquids willi successively lower boiling points but the principle is
the same as explained above ; (2) the Limle and Haropsons methods
employing the JouleThomson effect; (3) the Claude and Heylandt
methods which utilise the cooling produced when a gas expands doing
♦Michael Faraday (17911867), the "prince of experimenters, was bora of
htunbto i  to London. At the age of thirteen he became errand boy to a
•.Uer but later in 1813 he got employment under Sir Humphrey Davy at
the Royal Institution where be carried on his scientific work and finally succeeded
Davy in 1827 as Director of the Royal Institution. His greatest work ls the dis
covery of electromagnetic induction in 1831.
VI.]
SERIES KIlFRIGERATION
135
external work. We shall now consider these methods in greater
detail. 01 these the first is historically the oldest and theoretsqjdlv
the most efficient hut is somewhat cumbersome and is very little used
at present.
15* The Principle of Cascades or Series Refrigeration. — The method
was first employed by R. Pictet in 1878. In principle it may be
described as a number of compression machines in series. Pictel
employed machines containing sulphur dioxide and carbon dioxide
and obtained temporarily a jet of liquid oxygen by allowing com
pressed oxygen to expand adiabatically. Wroblewski and Olszewski
at Cracow obtained sufficient quantities of liquid oxygen, nitrogen
and carbon monoxide by the cascade method and determined their
properties. Olszewski used ethylene as another intermediary below
carbon dioxide and could thereby cool these gases below their critical
temperature. Kamerlingh Onnes later employed die combination
of methyl chloride and ethylene for liquefying oxygen. The prin
ciple of the method is illustrated in Fig. 1. Machine (1) utilizes
methyl chloride. This has got a critical temperature of M3 C C and
hence at room temperature it can be easily liquefied by the pressure
of a few atmospheres only. Water at room temperature flows in the
liquid
[lea, a
Fig. 7. — Illustration of the Method of Cascades.
ket in (1). Liquid CH 3 C1 falls irrto the jacket in (2) which is
connected to the suction side of the compression pump. Thus the
Liquid evaporates under reduced pressure and its temperature falls to
• hi 90 Q C. The compressor returns compressed CH 3 CI gas through
the tube inside jacket (1), which is shown straight but is really in
the form of spirals.
Inside the jacket (2) is placed the condenser coil through which
ethylene passes from the compressor or the gas cylinder. It liquefies
and i!i. . enters the jacket (3). There it evaporates under reduced
PRODUCTION OK LOW TliMPJERATURES
[CKAP.
ire and lowers Che temperaAiwre to about  I60°C. Tbj
tube insi acket (3) oxygen from a and liq
essure. Liqu ollected In Ehe sfe D.
The lowest temp;. obtained by boilin en under redt
pressure is 218°C which is higher than tl matures of
(228.7), hydrogen (240*0) and helium, hence the method
of cascades failed to liquefy these three gases.
The method of cascades is very useful for laboratory purpc
The use of the compressor can be entirely dispensed with by the use
uable liquids boiling under atmospheric or red essure.
: suitable liquids can be selected from Table . res the
normal boiling point, the critical temperature and the tri int Cor
the common gases. The interval between die critical temperature
and the triple poirj icnts the range in which the liquid is avail
able. Still lower temperatures ran be obtained by further reducing
the press. ■ the liquid when it solidifies. Thus, using solid
nitrogen and a good pump a temperature of about 2 an be
obtaim :] above the critical temperature qf neon and it
is not possible to bridge the gaps a nitrogen and neon and
between hydrogen and' helium in this way.
■
Table 2— G '", normal boiling point
and tri ases
Sul
B. P, at 1 atm.
Critical tem
Triple point
pres s
per:: .
Mechvl chloride
 24.09°C
143.3
 10S.9 C C
Sulphur dioxide
 10.1
157
lOnia
131.9
77.7
Car; :ide
78.6
31.0
56.6
Nitric oxide
89,8
36,50
 102,3
Ethylene
in .
9,50
169
ane
161.37
2,85
183.15
Carbon monoxide .
.1900
13
Oxygen
 182.95
_m
218.4
Nitrogen
19!
W
209,86
.
245
228.71
2'
Hydro
252
239.9?
259.14
Helium
267
16. Production of Low Temperature* by utilizing the JonleThomcon
Effect. — As the above method is not capable of liquefying hydrogen
and helium, another process began to be utilized from 1898, This
Is the JouleThorason effect discovered in 1852. A fall mathem
VI.
THEORY OF POROUS riAIG EXPERIMENT
187
analysis of the phenomenon is postponed to Chapter X. We
describe the phfnomem n hare:
We have already described Joule's eriment (p. which
showed, that for permaneja the internal em tot depend
upon volume, "I.e., tjp  = 0. This is called Joule's law or Mayer's
hypothesis and is the characteristic property of a perfect gas (for
perfect monatomic gases U — ^NkT per rnoIA But Lids is not strictly
true [or idle actual gases of nature ; they all show deviations from the
st: te of perfectness and hence for them U is not independent of
e. A slight change in the temperature of the gas should occur
in joule's experiment, but since the capacity for heat of the contained
air" is negligible as compared to the heat capacity of the surrounding
, no change in temperature could be observed. In 1852 Lord
Kelvin, in collaboration with Joule, devised a modification or Joule's
experiment in which very small changes in temperature produced by
expansion could be easily measured. This is called the "Porous plus
experiment and provi an unfailing test of Mayer's hypoi
Widi its help we can easily find how far a gas deviates from the state
of being perfect. Before proceeding to describe this experiment we
shall discuss the theory underlying it.
17. Elementary Theory of the Porous Plug Experiment.— In this
experiment a highly compressed gas is being continuously forced at
a constant pressure through a constricted nozzle or porous plug. The
as its name implies", consists simply of a porous material, say,
cottonwool, silk, etc. having a number of fine holes or pores u
thus equivalent to a number of narrow orifices in parallel. TT
during its passage through tlie pores becomes throttled or wiredrawn,
viz., molecules of the gas are drawn further apart from one another
doing internal work against molecular attractions. This is always the
case whenever a fluid has to escape through a partly obstructed
passage. On either side of the plug constant pressures are maintained,
the pressure on the side from which the gas flows being much greater
than on the other side since the plug offers great resistance to the
flow of the gas. This expansion is of a character essentially different
from Joule's e ision. In Joule expansion, the gas expanded with
out doing any external work. Here it expands against a constant
externa! pressure and hence it has to do external work also, together
with any internal work, while some work is done on the gas as well,
plug is surrounded by a nonconducting jacket so that the process
is adiabatic in the sense that no ""heat enters or leaves the system.
For such 'processes we now proceed to show that the total heat function
h = u { pv remains constant.
To prove this theorem, let us consider a. mass of the fluid
ous plug C from left to right as indicated in Fi
.. ,.. 7" A , u A and p E} v Bf T B , u a be the pressure, volume,
temperature and internal energy of one gram of the fluid before and
after traversing the en":; re: respectively. Suppose that one gram of
138
PRODUCTION OF LOW TEMPERATURES
I CHAP.
L
Vh
C
•/.:
M
the gas is contained between the porous plug and some point.
M on the left and also between
the plug unci some point N on the
right For visualising the process
we may assume an imaginary
piston A at M separating this
quantity of the gas, and that the
flow of the gas is caused by the
{or ward motion of the piston A.
Actually, however, the rest of the
gas exerts a pressure p± at M
which is maintained by the source
of supply. The gas after travers
ing the plug pushes forward the
imaginary piston B whose motion
is opposed by the pressure p of
the gas to the right of B. The
initial and final states are she
respectively at (a) and (b) in the
figure, the initial volume of the
Fig. &,— The Porous Plug
Experiment.
.gas being equal to ML and the final volume equal to ON,
The gas during its passage through the orifices in the plug has
to overcome friction, viscosity, etc., and hence loses energy. The
escaping gas issues in the form of eddies and its temperature falls
cor ily just at the jet (and this effect is spurious) because some
th'. i rgy is now converted to tile energy of mass motion. The
x, subside after traversing a short distance and the
.i.ture consequently rises. Let us consider only the steady
and after transmission through the orifice, i.*?.., at points
he plug where eddies are not present. We assume
lowly so that the energy of mass motion
is very sural! and negligible in comparison t nergy of thermal
motions. Now our initial system is ML and the final system is ON.
But the plug OL is initially and finally in the same state ; hence
the change simply consists in a change from MO to LN* If in
addition, the tube is surrounded by a nonconducting material, no
heat is supplied to or withdrawn from the system. Some work is,
however, done by the external forces on the slowly moving system.
The force acting at A is equal to p A )< S where S is the crosssectional
area of the cylinder. The work done by this force upon the
i: ' ;' } a)< 5 X MO = p x v A . Similarly, the work done by the gas in forcing
the piston B is p^v*. Therefore, the net work done by the gas is
put's — PaV*> and from the first law of thermodynamics, since
A(> — 0, tile work done by the system is equal to the decrease in
its internal energy, viz.,
»a "b=Mb^Ma (i)
OT *A +J*A»A = »S+J&B»B .. (2)
Hence u f pv remains constant in the throttling process.
"0
THE POROL'S PLUG EXPERIMENT
139
For perfect gases Boyle's law flw) T ^ constant and Joules law
u^=cT hold true, and therefore u\pv would depend upon tempera
cure only. We have just shown that u+pv for the porous plug experi
ment is constant whether the gas is perfect or not. Hence, tor
perfect oases the temperature on both sides of the plug would remain
the same. In actual experiments, however, a cooling effect was
observed £or most gases such as air, 2 , N 2 , CO, and a heating effect
Thus none of the gases examined was peiletf. inc.
be due either to deviations from Boyle s
both* If we know the former,
(p. 97) we am find the latter
experiment. If at the particular
pressure and temperature 1 1 A, the gas is more compressi
than
<*);.
alone
;as is
Upon
from
work
in case of Hj.
lack of per redness may
law or from Joule's law or
as from Amagat's experiments
by performing the porous^ plug
pressure and temperature in A.
at lower pressures [c£, curves of N* C0 2 betore the bend (
& z?* <PuVn. Therefore, due to deviations from Boyle s k
ut <ii ie. the gas would show a cooling effect*. IT the
l?ss compressible [cf., H a ) there would be a heating effect,
these effects will be superposed, the effect due to deviation
Toule's law. Since in actual gases, cohesive forces are present
will be done in drawing the molecules further apart during expansion
and the gas would become cooled. Thus the jouleThomson effect
due to this cause would always be a cooling effect. Ihe observed
effect is the resultant of these' two effects and may be a heating or
cooling effect depending upon the sign and magnitude of the former
effect/
18. The Porous Plug Experiment,— We shall now describe
actual experiment. Joulef and Thomson
were the first to carry out these experiments.
They employed a cylindrical plug, which is
indicated in' Fig, 9. The compressed _ gas
Hows through a copper spiral immersed in a
thermostat and after having acquired its
temperature, passes through the porous plug
W* The plug consists of silk, or cottonwool
or other porous material, kept in position
between two pieces of wiregauze and enclosed
in a cylinder of some nonconducting wood
bb. The plug and part of the tube is sur
rounded by asbestos contained in a tin cylin
der zz so that no heat reaches it from the bath.
Joule and Thomson worked with air, 2 , N 2 .
CO,, between 4° and 100°C, the initial and
final pressures being 4.5 atmospheres and 1
atmosphere respectively.
JoaleTnom
soh's porous pirns;
of two
energy
'♦It is important to remember that the internal energy is made up
parts ' ) kinetic energy which depends upon the temperature, (..• 1
due to molecular attractions, the former being much greater in a gas uik
ordinary conditions so that Joule's law is approximately obeyed.
{ foalej Scientific Papers, Vol. II, p. 216.
140
PRODUCTION OF 1X.AV TEMPtiRAi
CHAP.
Some of the subsequent worl^ers employed a plug of the "axial
flow" type as used by Joule and Thomson, while some others
err, i plug of the "radial How" type. In the latter the gas
flowed from the outer side of a hollow cylindrical plug to the interior
and hence heat insulation was better. Certain others employed only
a throttle valve or a restricted orifice.
From these experiments Joule and Thomson found that the fall
in temperature was proportional to the difference in pressure on the
two sides of the plug, i.e., bd—k(}> A —f! u ), where k is a constant,
characteristic of the fluid. They found empirically that k — A/T*
where T h the absolute temperature of the gas, while RossInnes
found k = A I B/T, Hoxton, however, found that his results were
represented by the form.
£_.4+j'~+^j
(*}
:]?0
We can find a value of k from theoretical considerations i
Chap. X) , If the gas be supposed to obey van der Walls' equation
of state, it. can be shown diat the JouleThomson ef r
Exercise .[.— U; dues of a = 136 X MP atm  cm * an d & =
320 c.r. for a grammolecule gen at N.T.P. and C* = 703 cal.
akulate the JouleThomson effect from equation (i) ,
9 = I_ 0 1 x 10* A
Ap 7.03x4.18xl0 7 L 8< ^73
; 4. 18x10' d
01 X 10* _dcg.
7*03x4.18x10* atrm/cms"
1 r a van der Waals' gas the cooling
duced in the JouleThomson process.
I 11 * •' by the gas— 1 * ^dV = — — — ,
, • . Net work (external [ internal) done by the gas
pro
"
=^f7"ir 1 )+A( J fr i  A ) +
::■■■
2a
since from van der Waals' equation /?F — RT 4bp — (a/V) approx
lor the two sides of the plug.
will produce a cooling by _ &q (since AC?  0) such that
(see Chap. X, sec. 10) 
VL]
PRINCIPLE OK KPGENK.RATIVK COOLING
14?
Wb'jt
f ^fe
or C B AG  #A0+*AH£j= (—Art a PP
Combining this i b on C„  C B t= R, we get equation (4) .
19, Principle of Regenerative CooUns— The JouleThomson «
in s observed tor most gases is v. as, £<w a jr at a tem "
oerature 20°C when the p iun on the two s.des are SU
atmospheres and 1 atmosphere respectively, Joule and Thomson found
that" ike tei;. re falls by IlVC. Hence the method eduid not
be employed for a long time tor producing liquefaction. Subsequent
however, it was discovered that the cooling effect can be unengi!
by employing what is called the reg ■ A portion ol
tne eas which first suffers JouleThomson expansion and becomes
mp] wed to cool other portions of the incoming gas betore
die la bes the nozzle. The incoming gas becomes still m *
cooled after traversing the nozzle. In this way the cooling eltect
can be made cumulative. In actual practice* this is secured by using
either concentric tubes as in Linde's process or by means of Hampson
spii r : i . Two or more concentric tubes are arranged in tlie
form of spirals, die inner one carrying the highpressure inflow
gas while die outer one die lowpressure outflowing gas. In the
regenerative method a further advantage
is gained by the fact that the lower the
ture the greater is the Joule
Thomson cooling.
The regenerative principle is illustra
ted diagram matically in Fig. 10 where the
high pressure gas from the compi
enters the spirals contained in the water
cooled jacket A. The gas next enters the
regenerator coils at E and by expansion at
the valve C becomes cooled by a small
amount. This returns by ,thc outer tube
abstracting heat 'from the high pressure
gas, and reaches F almost at the
temperature as the incoming gas at E.
The As is again compressed and cooled by
A and reenters at E. As time passes, the
gas approaching C becomes cooled more
and more till the JouleThomson cooling
at C is sufficient to liquefy it viz., its
temperature reaches the value at which
the gas would liquefy under the pressure prevailing at F. A portion
of the esc::: i j gas then condenses inside the Dewar flask D. At
this stage the temperature throughout the apparatus becomes steady
i . ic represented by the curve shown in Fig, 11, p. 142. The
part L M" represents the continuous decrease of temperature of the
as we approach the nozzle through the inner tube while MN repre
sents die JouleThomson cooling. NL represents the temperature
of the lowpressure gas which is less than that of the adjacent high
f :i; i ;l
Fig. 10. — Illustration of the
Regenerative Coding.
142
i'JiODUCTION OF LOW TEMPERATURES
CHAP,
pressure gas. Thus the cooler lowpressure gas abstracts heat from
die incoming stream.
20. Linde's Machine for
liquefying Air,— The principle
of regeneration (applied to heat
ing) was discovered by W. Sie
mens in 1*57, hut its application
to cooling' processes came later.
Linde* in Germany and Hamp
son in England almost simulta
neously (1895) built air lique
fying machines based on the
above priii rip J e. At the present
time, such machines have be
come quite common and many
ies in tire world arc fitted up with such machines. Fig, 12
3 a commercial form of Linde's machine; e, d, is a two, or better
]}Js!anc« along (he Mgciicmlve tail
1ml\ II, — Temperature distribution
1 ... 12.~Linde's apparatus for liquefying air,
three stage compressor, the machine e compressing the gas from 1 to
20 atmospheres while d compresses it from 20 to 200 atm, A charge
oJ atmospheric air is taken in at. e and compressed by e to 20 atm.
* K:ir ' y» Linde, born in 1842, was Professor at Munich. He published an
account of bis air 1 . machine in 1895.
VI.]
AIR LIQtttiFIERS
143
then cooled by passage through watercooled tubes and is ^
to the suction side of the second stage compressor. The tomprebsed
L d!en pa^es tough the cylinder f which contains caustic soda.
J his a^o?b tie carbSn dioxide (IE this is not done, carbon dioxide
ifflbSS? Nidified and choke the valves in the ^^Hggg;
■ as then pusses to the tubes g which are cooled by a freeing
m ixttne to . 20'C, From here it, passes on through the metallic tubw
p^hiner coils of the liquefier proper. At a we have got the plug
<in the first stage to 20 aim.), tb* temperature falls to about <B <£
ind the air agali passes through the outer cods cooling the incoming
t» it is then led Sirowh the pipe P, to the second stage compression
fnder where it is again compressed and allowed to pass through
X reftigerator and thl inner coils to a. After the ^p etion o a
few cycles the temperature of the incoming gas falls so low that the
xerond throttle valve is opened The air .s now allowed to exp tndto
1 atmosphere when it becomes liquid and ^ ct %^j^^ .:
from which it can be removed by the siphon A. The unliquefied gas
is aeain led back through the outermost cods to the compressor e as
nSd by the arrowt Fresh charge of air is beifg continuou
ta^en in at*, compressed and delivered along with the gas from the
middle tube to d. The process is cyclic.
21. Hampso^s Air Liquefier. This liquefier also lujlizes the
jouleThomson effect and the regenerative principle, but differs i from
Linde's apparatus in details of construction. The special feature
about it is the Hampton spiral. The high pressure inflowing ^as
Passes through copper* tubes coiled in the form of concentric spirals
arranged in layers; it then suffers the throttle expansion and becomes
cooled This cooled air vises through the interstices between the
a era or spiral and thereby cools the incoming li^hpr^iire gas.
After some time the highpressure gas becomes sufficiently cooled so
that on suffering the throttle expansion, it liquefies. The apparatus
thus differs from that of Lmde only in the manner of cooling the m
comine 2as. The apparatus was later improved bv Olszewski. 1 he
ITampson construction has been utilized later by Dewar, Onnes and
Meissner for liquefying hydrogen (see Fig. 14) and helium.
22. Claud* Air Liquefier* Although the Linde and Hamp:
lynchers just described are in extensive use in laboratories and com
mercial installations, the machines cannot be said to be satislactoi
ainly because the efficiency of the machine, t.e. f beat extracted/
wiergy consumed bv the machine, is extremely low (about 15%). llie
coolinir process employing JouleThomson expansion is really very
tent. A more efficient machine could be devised if the com
,. K as was made to expand adiabatically doing external work
and thereby suffered cooling, The technical difficulties in construct
,i apparatus for continuous liquefaction of gases by adiabatic
expansion were overcome by Claude. The main difficulty consists
in finding a suitable lubricant for the moving parts of the expan
sion cvlinder since the ordinary lubricants become solidified at these
m
en
144
PRODUCTION OF LOW TEMPERATURES
[CHAP.
low temperatures.
C
d) Low Pressure Air
Fig. 13. ClaudeHeylaiidt s;.
laude utilized petroleum ether as the lubricant.
This ... i . iscous at tempera
Canvm* tures of  140*C, or even  160*C
and thus acts as an effective lubri
cant up to this range.
13 shows diag i i
the Claude's air liquefying:
chine, The gas (ram the com
5 divided Into two parts
to the expau
cylinder and suffers adiabatie
onsequent coo]
In this expansion it does
work which is utilized in doing
work on the compressor. The
:d gas traverses upwards in ihe
pipe D thereby coi i he second
n of the Incoming com :
in the second heatexchanger.
The highpressure gas thus parti
ally liquefies. It then suffers Joule
Thomson expansion at the throttle' valve. The evaporated r
taken to the compressor and afi
Theoretically ( I ulc * be more efficient than the
only slightly so. This is
ilties of • ide method, The
1 has the great advantage that no movable parts of the
ad hence the csmtructia
dt slight Lqucfier.
U ; taiuion turbine will po vera!
dprocariug engine, Kapitza in 1939 developed
ier in which the compressed air is allowed to expand
ire of abc ; ressure of about 15 atmos,
and drive a turbine wheel, and thereby suffer cooling on account of
ie. The machine is about three times as efficient as
Lmde's li is it utilizes a pressure of about 5 atmospheres
only, all danger due to high pressure is eliminated.
23. Liquefaction of Hydrogen.— The method of cascades failed*
to liquefy hydrogen, Wrohlewski thereupon studied the isotherms
i gen at low temperatures, ai that caL the values
and b, and thence the critical constants (p. 95) . The critical
1. unci to be very low (— 240°C.) and Olszewski
, ted out that this temperature could not be reached by the evapo
n of liquid nitrogen, which was the most intern
shen known. The physicists then turned to the JouleThomson
This !• however, first appeared to be inapplicable to
Regnault had shown that when hydrogen
♦Neon can be utilised but it is very rare.
VI.
HYDROGEN LIQUEFYING APPARATUS
H5
is subjected to this process, it gets heated instead of being cooled.
That the difficulty is not fundamental and insuperable can be seen
from the expression for the JouleThomson effect viz. }
• . ■ ■ (4)
ApCART }
It is thus proportional to 2a/RTb but in hydrogen and helium
a is so small that at ordinary temperatures 2a/RT<,bj and the term on
the righthand side becomes negative. Hence, for a negative value
of Aft *"■*» expansion of the gas, &Q is positive at ordinary tempera
tures, and the gas shows a heating effect.
If T be sufficiently reduced, the righthand term in (4) eventually
becomes positive and the gas shows a cooling effect. There is just
a temperature where 2a/R.t b — Q, i\fi v where JouleThomson effect
changes sign j this temperature is called the 'temperature of inversion'
T, which is thus equal to 2a/ bR. This relation gives, after substi
tuting the values of a and b for hydrogen, T, to be  73 1> C Olszewski
experimentally observed the JouleThomson effect for hydrogen at
various temperatures and found the temperature of inversion to
be  80.5 fl C.
Similar is the case with all gases. There is a temperature of
inversion for all of them which, however, depends upon the initial
pressure of the gas. Even the JouleThomson effect, depends very
much upon the initial pressure. It is thus dear that the behaviour
of hydrogen and helium is not anomalous ; they differ from other
gases only in having a low temperature of inversion.
Now since T t » the critical temperature, is equal to da/27hR
(p, 95) and Ti = 2a/bR f it follows that T i= (27/4) T t . This rela
tion is found to hold true approximately.
Hydrogen must, therefore, be cooled below ~80 D C, for liquefac
tion. But "for practical success it should be precooled to the Boyle
point T B (p, 88) which is defined as the point at which — ^— — 0.
Calculation with the help of van der Waals" equation shows that
this temperature T B =a/bR r Hence r B = £7V We thus see that
hydrogen should be precooled to about 96°K { 177'C). This tem
perature in easily attained if we immerse the hydrogen liquefying
apparatus in a bath of liquid air.
24. Hydrogen Liquefying Apparatus. — Dewar first succeeded in
liquefying hydrogen in tliis manner in the year 1898, Travers later
i 1 1 1 1 i : /red the apparatus. Hydrogen prepared from zinc and sulphuric
acid is compressed to about 150 atrn, and then passed through coils
immersed in water in order to deprive the gas of the heat
Of compression. Next it passes through cylinders of caustic potash
dehydrating agent and is deprived of its carbon dioxide and
moisture. This is essential as these impurities would solidify much
before the liquefaction of hydrogen sets in and choke the tubes. The
10
146
l'RQDUCTION OF LOW TEMPERA! L RES
[CHAP,
gas then enters the liquefier and traverses the regenerative coils A
(Fig. 13) which are cooled by the
outgoing cold hydrogen gas, and in
the final steady state, becomes
cooled to about — I70°G. Next
the gas passes through a refrigerat .
ing coil B immersed in liquid air, and
then through another refrigerat : n .;
coil C immersed in liquid air boil
ing at a pressure of 100 mm. This
is adjusted by allowing liquid air
from F to trickle into G and by eva
cuating G through a pump attached
ai r. The tempera tine of the iiv
drogen gas thus falls to about
200°C. Alter this it traverses the
coil D and suffers JouleThomson
expansion at the valve a which is
operated by H. The gas thus be
comes cooled and this cold gas
passes up round the chambers G
and F, thereby cooling the coils D
and C, to the chamber R and from
there to the compressor. Thus ■
alter a few cycles the temperature
of the incoming gas at a falls to
25Q D C, and then on suffering the
JouleThomson expansion it liquefies
and drops as liquid into the Dewar
! I V.
Fig. 14. — Hydrogen liqw
Many later investigators devised apparatus which have a large
output. Amongst them may be mentioned Nernst, Kamerlingh
Onncs and Meissner. Dimes' apparatus h very similar to his helium
liquefier. Meissner's apparatus is somewhat different in construction
but similar in principle.
Liquid hydrogen boils at 252.78 C C, under atmospheric pressure.
By causing it to boil under reduced pressure it can be frozen to a
white solid.
25. Liquefaction of Helium. — Helium could not be liquefied
for a long time. The attempts of Dewar and Olszewski to liquefy
helium by the adiabatic expansion method were unsuccessful
Kamerlingh Onnes,* however, proceeded in his efforts very systemati
cally. He studied the isotherms of helium down to liquid hydrogen
temperatures (up to— 250°C) and obtained the critical constants for
helium. He found the following values :—T { = 5.25'K, p ( = 2.26
aim. and normal boiling point — "4.2f>°K. The jouleThomson jnver
*He£ke Kamerlaiifih On ( 53— — 1926), bnrn in Holland, became Professor
of "Physics at Leiden where he established his low temperature laboratory and
investigated the properties, of substances at low temp
vi.J
SOLIDIFICATION OF HELIUM
M7
sion point came out to be about •55°K. and the Boyle point 17 :: K.
is temperature could, therefore, be readied by preeooling the gas
with liquid hydrogen. Kamerlingh Onnes: was thus convinced of the
possibility of being able to liquet)' helium by the Linde process. He
u actually liquefying it in 1908 in his laboratory ac Leiden.
Subsequently helium liquefiers were constructed at Leiden, Berlin
and Toronto. Today there are scores of helium liquefiers in the
world. Since helium is rather costly, the arrangement should be
such that it can work in cycles, In the apparatus used at the
t pyogenic laboratory at Leiden gaseous helium compressed to 36 atm.
is passed through spirals immersed in liquid hydrogen boiling under
reduced pressure and then through outgoing cold helium vapour.
The gas then suffers JouleThomson expansion and becomes liquefied.
The plant for liquefying helium is, therefore, complicated by arrange
ments tor liquefying air and hydrogen.
Both hydrogen and helium were liquefied by Kapitza in 1934 by
the ClaudeHeylandt method. Helium was liquefied by Simon by the
adiabatic expansion method and also by the desorption method.
Collins in 1917 developed a commercial type of helium liquefier based
on the Kapitza method. In the Collins expansion engine the piston
and cylinder are constructed of nitrided nitralioy steel, the clearance
being about 0.0005 inches on the diameter and the operation being
completely dry. Thus the leakage of gas is extremely small and
whatever does leak, also goes to the suction side of the compressor.
26. Solidification of Helium. — Kamerlingh Onnes tried to solidify
helium by boiling it under reduced pressure, but though he claimed
to have reached 1.15 n K in 1910, helium still remained a fluid. In
1921 he again tackled the problem and by employing a battery of
large diffusion pumps he reduced the vapour pressure to 0.01.3 mm.
and the temperature to 0.31°K,but helium still remained fluid. After
the death of Onnes, his collaborator and successor, Dr. Keesoni suc
ceeded in 1926 in solidifying helium by subjecting it to an enormous
pressure, Helium was compressed in a narrow brass tube under a
fressure of 130 atmospheres, the tube itself being immersed in a
[quid helium bath. Jt was found that the tube was blocked indicat
• that part of the gas had solidified. If the pressure was reduced
I or 2 atmospheres, the tube became clear again. Later expen
ds showed that helium at 4.2*K solidified at 140 atmospheres
while at 1.1 °K it solidified only under 23 atmospheres. Solid helium
i be distinguished from the liquid : it is a transparent mass
having almost the same refractive index as the liquid.
27. Cooling produced by Adiabatic Demagnetisation. — Upto 1925
tiethod available for producing temperature lower th
is the boiling of liquid helium under reduced pressure.
mi in this way reached 0.72°K in 1932, In 192G Debye and
u theoretically that lower temperatures could be pro
by the adiabatic demagnetisation of paramagnetic substances
(i.e. those substan ces for which the magnetic susceptibility y
' i i . The principle of the method is as follows ; —
M8
PRODUCTION OF LOW TEMPERATURES
[CHAP.
The process of magnetising a substance involves doing work on
it in aligning the elementary magnets in the direction of the external
field. If a substance already maguc sei demagnetised adia
batically, it has to do work and the energy to do this work is drawn
from within itself, in consequence of which it cools, This cooling
can be made large if a .strong magnetic held is employed and the
initial temperature is low because then the magnetisation produced
in the substance is large. This follows from Curie';; law. which states
that the paramagnetic susceptibility of a substance varies inversely as
i u absolute temperature i.e. % = C/l\ The final temperature attain
ed is determined by measuring the magnetic susceptibility of the sub
stance and calculating from Curie's law. In this way de Haas and
Wiersma succeeded in reaching the present low temperature record
of about (UK)34° K in 198 > idiabatkally demagnetising mixed
crystals of chromiumpotaawum alum and aluminiumpotassium alum
at' 1.29°K from an initial field of 24000 gauss.
28. Properties of substances at liquid helium temperatures. —
Properties of substances undergo very interesting changes at ex
tremely low temperatures. K. Onnes found in 1911 that at abom
metals appear to lose completely their electrical i
and become superconducting. The resistance does not absolute
vanish bin falls bo aboui a millionth of its value so that if a current.
be induced in a toil ol Lhe metal placed inside such a low ten
bath by bringing a magnet near it, the current does not in
die out as in ordinary ele« ti aetic induction, but may continue to
together. It. has also been found thai near the absi
to vanish. Besides, liquid helium
about 2 esses strange properties, the most im
portant being the property of superfluidity when the liquid has no
viscosity.
29. Low Temperature
Technique* — Dewar ft.i
The discovery of Dewar flask
by Sir James Dewar* in the
Royal institution of Loi
provided a very convenient
apparatus for low tempera
ture storage. Though very
low temperatures had been
produced, it was d::
to maintain the liquids
at these temperatures as
even by packing the bottles
with the I:
material, the leakage oE heat, from outside could not be prevented.
But the problem • ed by Dewar in a very ingenious way,
* James Dewar (18421523), born at Kincardine, became Pn iess c oJ b
Philosophy at Cambridge In 18/5 n,nd also Pi • n the Royal
Institution in 1877, His i rfc was in the low temperature region,
15,— Dewar Flask,
VI.]
USES OF LIQUID AIR
149
The Dewar flask (shown in Fig. 15 together with a siphon) con
sists of a doublewalled glass vessel, the inside walls being silvered.
The air is completely evacuated from the interspace between the walls
which is then sealeti If some substance be now placed inside such
a vessel and the top closed, it is perfectly heat insulated, except for
the small amount of heat which may creep in by conduction alone
the sides. The silver coatings protect the inside from radiation, and
the absence of air prevents the passage of heat by conduction through
the walls. If the substance be hung by thin wires inside the flask
and i,l le latter evacuated by pump and sealed, the insulation is com
plete. Such an arrangement was used by Nernst in his low tem
perature calorimetry (p. 42) . Dewar flasks are now sold in the
market under the trade name "Thermos flask". They have lately
been made entirely of metal with a long neck of some badly con
ducting alloy as (ierman silver.
Low Temperature Siphons. — For transferring liquid air from one
vessel to another, special types of siphons are used. One such siphon
is shown in Fig. 15, connected to the Dewar flask. It is formed of
a doublewalled tube silvered inside, the space between the walls
being evacuated. On the application of gentle pri • sure to the rubber
compressors A or B, liquid air rises up trie siphon and can be trans
ferred to a second vessel.
tvyostats. — For low temperature work constant temperature
baths are necessary; they are called cryostats. The substance to be
investigated is kept immersed 1 in these baths. From Table 3, (p. 136)
it is easy to find out which Liquids are suitable in a particular range
of temperatures. In this way suitable liquid baths can be easily
constructed down to ~218°C/ When no suitable liquids are avail
able, vapours of liquids can be employed.
30. Uses of Liquid Air and Other Liquefied Gases. — The import
ance of liquid air is being increasingly felt so much so that it has
now become essential for several purposes. Bottles of liquid air
can now be obtained in any important modern town at a compara
i ill cost. We shall give some of the important uses to which
liquid air has been put r
(') Prod of High Vacuum, — High vacuum can be obtained
by using Liquefied gases with or without charcoal. For instance, if a
1 1 1 fust fdled with a less volatile gas than air, say sulphurous
i' id h water vapour, and is then surrounded by liquid air all the gas
inside becomes solidified and thus high vacuum is produced. If the
I contains air, liquid hydrogen may be employed to condense
ii. I Ilia process is greatly assisted by charcoal which possesses the
rem property of occluding gases at very low temperatures and
the i ; temperature the greater is the adsorption. Further also
•he ion is selective; as a general rule it may be said that the
more vola lie the gas, the less it is adsorbed.
shall give a numerical example. During a certain experi
i' i in a vessel containing air at a pressure of 1.7 mm. at 15 a C, when
l'UODUCTIOX OF LOW TEMPERA!!: •:.;•
r.i.\'. : 
cooled by charcoal immersed in liquid air, gave a pressure of 0.000047
mm. in an hour and using liquid hydrogen as the i oak sure
was reduced to 0.0000058 mm.
(ii) Analytical Uses of Air*— Liquid air is of great use in drying
and purifying gases. Water vapour and the less volatile impurities
are easily removed by surrounding the gas in question (say H>) with
liquid air, and for this purpose it is now used as a common labora
tory rea,:y
(Hi) Preparation of Gases from Liquid Alr.~ Oxygt a k now pre
pared commercially from liquid air by fractional distillation. Since
the boiling point of nitrogen is  195,8 C C, and that of oxygen is
 182,9°*!, the fraction to evaporate first will be rich in nitrogen while
thai evaporating last will be rich in oxygen. A few fractional distilla
tions will suffice to separate these completely , Several r
v been devised to effect this separation. In Linda's rectifier
(1902) liquid air trickles down a rectifying column where it meet, i
upgoing stream of gas. The temperature at the top of the column
is slightly below 194°C, {B, P. of liquid air) while at the bottom
it is  183 °C. (B. P. of oxygen). 'Ihe rising gas at the bottom
comes in contact with the downcoming liquid and thereby some
oxygen of tire risinu condensed, while some of the nitrogen in
the downcoming liquid evaporates, and the liquid also becomes
oner. The process continues till the liquid reaches the bottom
whi mains nearly pure oxygen, while nitrogen passes off as
mi at the top. gen is almost pure but die nitrogen con
of oxyg <re efficient rectifiers have since been
devised by other workers. For details see Separation of Gases by
. Run em an n, Chaps, VI, VII and VIII.
gain atmospheric air may be utilised for the production of the
rare gases, particularly helium, neon and argon. Roughly five
volumes of helium are found in million volumes of air but this is
sufficient for our purpose. Liquid air may be separated into two
fractions, the less volatile part consisting of O a , N 2 , A, C0 2 , Kr, Xe
and the more volatile part consisting of He, H 2f Ne.^ Thus in the
rectifier described above the gas going up will contain N tt , LL, He
and "N T e. The nitrogen is removed by passing the gas through a
dephlegmator and hydrogen is removed by sparking with oxygen.
Neon and helium can be separated by cooling the mixture with liquid
hydrogen. Thus, oxygen, nitrogen and helium may be obtained from
. For details see Ruhcrnann, Separation of Gases, Chap. IX.
(iv) Calorimelric Application*. — Dcwar constructed calorimeti
of liquid air, oxygen and hydrogen. lie employed pure lead as the
heater, and the volume of the gas evaporated by the application of
this heat was measured. These calorimeters have the advantage
that a large quantity of gas is formed which makes it possible to
detect as. little as 1/S0O calories with liquid hydrogen. In this way
V1 1 PRINCIPLES OF AIR CONDITIONING
the specific heat of lead and other substances may be investigated
at low temperatures.
(v) Use of Liquid Gases in Scientific Research^Jhe extreme y
low temperatures which are now available to us by the use o£ liquid
and ifquid hydrogen have opened for the investigator a new and
vast field for research. Tins has made a liquid air plant essentia
,,,;,, modern laboratory. Most of the important OT^""*
matter have been investigated at low temperatures and have y, elded
X of farreaching importance. This has been extended even »
biological research, where it has been shown that bacteria as well as
their activity unimpaired even after exposure to liquid
air temperatures though a moderately high temperature is fatal.
(vi) Industrial Uses of Liquid G««s.— Liquid air is used com
mercially for the preparation of liquid oxygen as explained above.
For submarines anS aeroplanes it may be found useful to store liquid
air or liquid oxygen for respiration but the low thermodynamic effi
ciency inherent in the Uncle machine prevents the use of liquid air
oxygen "is, employed on a small scale for preparing expk
with powdered charcoal and detonated, it explodes with gieat
violence.
31. Principles of Airconditioning. The Comfort Ctort^The
casonal variations of temperature, humidity, etc, have marked effect
m PTowth, longevity and working efficiency of man. The seasonal
of the Near lead lis to change periodically our clothing, food
on gro
changes of the year lead .
and manner of living. But we can hardly cope adequately with the
variations unless we" can really control the weather changes within
our comfort limits as regards temperature, humidity and other factors.
The science of refrigeration, heating and ventilating devices have
rendered it possible to control the weather at least within the tour
wails of our room. This particular branch of study is known as the
science of airconditioning.
Complete airconditioning means the control of the following
rs : .
.Average comfort condition.
757 7 'T.
6065%.
25 75 ft./min.
at least 25% of total circulation.
I)
Temperature
Relative Humidity
Air movement
Introducing Fresh Air
Purification of Air
i ii ido tzing
[<<iv Li i >r Activating the Air.
Although apparently temperature seems to be the only guiding
, i,„ ; Lng, the relative humidity (r.h.) plays almost
ly imp 'le in the feeling of warmth. The same tempera
m
PRODUCrriON OF low temperature?
[chap.
ture condition, say 75°F, may make us feel either a bit too warm
or too chilly according as the r.h, is too high or too low. This is
because the humidity condition controls the evaporation from our
body and hence the abstraction of latent heat which gives rise to
the different warmth feeling. It is interesting to note thai, some
of the airconditioning plants in the tropics (Calcutta) do not employ
any heating device during winter but only humidify the atmosphere
by atomised spray of water. This is because the average winter
temperature of the place (about 70 — 75°F indoors) is not 'really too
low, but what makes us feel chilly is the low (40%) r.h. So we feci
quite comfortable only by raising the r.h. up to 60— 70%. In cold
countries, however, rooms are conditioned by electrical or steampipe
heating devices*
it is interesting to find that the comfort feeling is fairly critical,
that is to say, that individual variation does not go much oft' the
average. Elaborate experiment? have been performed by the Harvard
School of Public Health in collaboration with the American Society
of Heating and Ventilating Engineers on the average comfort feeling
under the different combinations of temperature, humidity and air
Dlty BULB TEMPERATURE
Fijr, 16.— The comfort chart.
velocities, etc. As a result of these experiments the comfort chart
(Fig. 16) is drawn in which the coordinates are tbe dry and the wet
bulb temperatures and lines of constant r.h. and comfort scales for
vi.j
PRINCIPLES OF AIRCONDI HOKING
153
cliU
summer and winter are also drawn. The chart shows that 98% ol
the people during summer would feel very comfortable at 71°F.
effective temperature. We shall see presently what effective tem
perature really means. It relates to the human feeling of warmth
under various combinations of temperature, r.h. and air velocity.
For example, this 7l°F effective temperature., may be obtained by
various combinations, such as 40% r.h., 78°F dry bulb, 62°F wet
bulb ; or 60% r.h., 75°F dry bulb, 60 D F wet bulb ; or 70% r.h., 74°F
dry bulb, 67 b F wet. bulb (all with air velocities 15—25 ft./min.) . So
we see that the term 'effective temperature' represents a new scale
which enables us to standardise the comfort feeling due to the various
combinations of dry bulb and wet bulb temperatures (which automa
ticalfv define r.h.) and air movements. The three combinations
(starting with 40%., 60% and 70% r.h.) as exemplified above would
give rise to a feeling as if the person were placed at a temperature
of 71°F in a saturated atmosphere and with still air. This is how
we can define effective temperature.
The effective temperature scale thus represents the conditions
of equal warmth feeling with various combinations of temperature,
humidity and air move ^
men is. These are find
ings based on experi
ments with men and
women with normal
clothing and activity
and subjected to condi
tioned atmospheres of
the various combina
tions of temperature,
r.h., and air movements.
Fig. 17 shows the
el i :: c rive tempera ture
chart. Let us see how
to read the chart to find
the effective tempera
ture. Suppose that we
have in trie room an
atmosphere with dry
bulb temperature 76°F,
and wet bulb 62 D F and
wind velocity "30 ft./
min r (030 ft./min, may be taken as good as still air.) Now put the
rht edge of a scale across the two temperatures (the dotted line) ,
and it intersects the effective temperature lines at about 70°F. Tf the
same temperature of 7G°F dry bulb and 62°F wet bulb are found to
exist with a wind velocity of 200 ft./min. our feeling would be corres
ponding to (574°F effective temperature and so on."
We have seen that temperature and humidity are the most
important factors for comfort feeling, while movement of air gives
17.— Effective temperature chart.
PRODUCTION OF LOW TEMPERATURES
CHAP.
a feeling of ease if ic has velocities within 2575 Et/min. If the
air is dead still, it becomes uncomfortable and stuffy. On the other
hand, if it has too high a velocity it. becomes blasty and we would
not like it. The higher the wind velocity, the colder is our feeling,
as it facilitates evaporation from our body.
We next consider fresh air. A roomcooler (or a roomheater
in the cold season) must make provision for introducing sufficient
amount of fresh air into the room. Of the total circulation a mini
mum of 25% fresh air is recommended, the remaining part is the
room air itself, recirculating through the machine. It h true that
it more air is drawn from outside and cooled in the airconditioning
machine for distribution in the room it would be better, but it becomes
too expensive since much more work has got to be done in ord
cool the large bulk of outside hot air.
In a complete airconditioning outfit, devices are included to
purify and deodorize the air by suitable means.
In spite of the complete airconditioning arrangements it is found
that we can never feel like the natural atmosphere under the same
conditions. Recently it has been found that the amount of electri
cally charge ions present in fresh atmosphere is higher than that
present in the air of an occupied room. Experiments have been
made to ionise air by Xrays and introduce the ions into the room in
portions to activate the atmosphere of the room. This has
given positive effect.
32. The AirConditioning Machine, — We have so far seen what
iing actually in: Ve shall now consider how it is
ved.
Fig. 18.— The Evaporator.
The airconditioning machine or the roomcooler is fundamentally
a refrigerating machine which has been described in section 7, p. 130
with the main difference in the design of its evaporators. This is
THE AlRCONDlTlONlNG MACHINE
155.
VI.]
just tl»e question of how we want to utilize the cold produced by the
refrigerating machine. In airconditioning unit the evaporator con
sists of a scries of zigzag coppertubings thoroughly firmed with thin
copper sheets in order to get a large area of cold surface (Fig. 18) 
As the liquefied refrigerant (SO*, Freon, etc.) evaporates in the tubing
fier, etc, clearly
HB&
i**Wl«'«B»r.*i.v:..
Fig. 19. — Frigidaire conditioner,
It is important, to note that die summer air is laden with much
moisture and it is desirable that humidity should be lowered. As
the air is fanned through the cold fins of the evaporator the moisture
condenses on them into droplets which are ultimately drained off.
Thus cooling and dehumidification are simultaneously brought about
in die same process.
156
PRODUCTION OF LOW TEMPERATURES
[CH IP.
The size and capacity of an airconditioning machine is not
determined only by the size of a room. It depends' upon the follow
ing considerations of heat loads ;
(i) Sun's rays falling on walls or roof.
(ii) Conduction through walls and roois due to the difference of
outside and inside temperatures,
(tti) Human occupancy. — For small private installations this heat
load is not more than 5% of the total load but in cinema or
theatre halls, it is 55 to 65% and in restaurant 40 to 60%.
(Average heat dissipation is taken to be 400 13. Lu. per hour
by each person) .
(iv) Infiltration, Le. } outside unconditioned air entering through
conditioning machine itself (its motor) , cooking stove, etc.
(v) Heatproducing items in the room, e.g. tie lamps, air
conditioning machine itself (its motor), cooking stove, etc.
In order to minimise the heat load which mostly enters from
outside, the walls and ceiling must be covered with insulating boards
such as celotcx, masonite, etc. and matting for the floor should
he used.
America and most of the European countries, But for tropical
like India, the chart would differ considerably. People in
the tropi omed to more warmth and humid atmosphere,
an{1 tni ican airconditioning machines have got to
readjusted according to our comfort conditions.
Books Recommend*
1. Andradi
Z I,. C. Jackson, Low Temperature Physics (1950),
Mediuen & Co.
3. M. and B. Ruhemann, Low Temperature Physics (l°§7) ,
Gam bridge University Press.
4. C. F. Squire, Low Temperature Physic* (1953) , McGrawHill
Book Co,
5. M. Ruhemann, The Separation of Gases (1940) , Clarendon
Press, Oxford
fi. Glaze brook, A Dictionary of Applied Physics, Vol 1, articles
on 'Refrigeration' and 'Liquefaction'.
7, Mover and Fittz, Refrigeration,
8, Hull, Household Refrigeration, Published bv Nickerson &
Collins Co., Chicago.
9, K. Mendelssohn, Cryophysics, (I960), Intersciencc Publishers,
Inc., New York.
CHAP PER VII
THERMAL EXPANSION
1, The size of all material bodies changes on being heated. In
the majority of cases, the size increases with rise in temperature,
the important exceptions being water and some aqueous solutions in
the range to 4 fi C, and the iodide Of silver (resolidified) below
142°C. We shall first consider the expansion of solids.
EXPANSION OF SOLIDS
2, The cubical expansion of solids is somewhat difficult to
measure directly (a method is given in section 17) , and is generally
calculated from the linear expansion. Hence, experiments on die
expansion of solids generally consist in measuring the linear expansion
of bars or rods of the solid. For isotropic bodies whose properties
are the same in all directions, the expansion is also the same in all
directions. To this class belong amorphous solids (e.g., glass) and
regular systems of crystals (e.g., rock salt). Metals may also be in
cluded because though they are composed of a very large number
of small crystals, these crystals are oriented at random and the
average properties are independent of direction. In anisotropic
bodies such as many crystals, the expansion is different in different
directions and may be even of different sign. We shall first consider
isotropic bodies.
ISOTROPIC SOLIDS
3. Linear Expansion. — If a bar of length I, } at 0°C. occupies a
length I: when raised to t a C. t l t can always be expressed by a rela
tion of the form
where X is called the mean coefficient of linear expansion between
and t°Q., and is a very small quantity. This differs very little
from the true coefficient of linear expansion « at the temperature
.' which is equal to
1 dl
I dt'
The true coefficient a may also be defined
] <!l
by the relation a=, — j~ which on integration will yield (1) if
to at
fadi — \l. The mean coefficient may be put equal to the coefficient
of expansion at f/2°C if the range of temperature is small. Often
the initial length is measured not at 0°C but at r_ L Q C. Then if t. z
denotes the other temperature at which the lengh is l& we have
158
THERMAL EXPANSION
[QIAP.
Hence
by the binomial expansion
A = 5S=S approx
The mean coefficient A itself is found to vary with temperature.
This implies that the relation connecting length and temperature is
not a linear one and equation (1) must be modified into
where the successive coefficients go on decreasing rapidly. An equa
tion of this type is entirely empirical. The molecular theory of matter
lias not yet been developed sufficiently to yield an exact 'theoretical
formula. Generally it is sufficient to include terms up to the square
of t ; the relation then becomes parabolic. In most cases both the
coefficients Aj and Aa are positive, the body becoming more expansible
as the temperature rises.
4. Earlier Measurements of Linear Expansion. — The Linear expan
sion of solids is very small ; a bar of iron one metre long when heated
from to 100°C, increases in length by about 1.2 mm." To measure
such small changes in length accurately, special devices are necessary.
The increase in length may be obtained from the readings of a sphero
m eter, or directly observed by means of a microscope. Again , the
expansion may be multiplied in a known ratio by utilising the principle
of the lever. The most satisfactory method, however, consists in
utilising the interference fringes, which is considered in detail later.
In this section, we shall consider the earlier experiments.
The spheroiriLtcr or a micrometer screw was generally employed
to measure the expansion and is suitable for ordinary work. The
f. about a metre long, has its one end pressed against
ted screw while the other end is free to expand. There is a micro
■ ■ spheroineter which can be brought into contact with
his em noting the micrometer readings when the screw is in
contact at 0°C., and at any other temperature (°C„ the expansion of
the rod is found, whence the mean coefficient of linear expansion can
be calculated from equation (1),
Roy and Ramsden employed microscopes to measure the cxpan
□ and were able to obtain results of considerable accuracy. The
experimental bar was placed horizontally in a trough between two
standard bars, and parallel to them. One standard bar carries a
crossrnark at each end while the other carries at either end an eyi
piece provided with crosswires. The experimental bar carries an
object glass at both ends so that the eyepiece on tire standard rod
and the object glass on the experimental bar together formed a
microscope focussed on the crossmark on the second standard b: ,
The standard bars were always kept in ice. One end of the experi
ntal bar was fixed while the other end was free to move when the
bar was heated. The object glass was brought back to its initial
position by a fine micrometer screw, whose initial and final readings
the expansion.
ru.J
COMPARATOR METHOD
159
1 . — Apparatus of
Lavoisier.
Laplace and Lavoisier employed the lever principle to magnify
the expansion (mechanical lever method) , The change in length was
converted into a change
in angle by means of a
lever arrangement and
the angular change was
measured by a scale and
a mirror or telescope. The
principle of their appara
tus is indicated in Fig. 1.
One end A of the experi
mental bar AB is fixed
while the other end B
pushes against a vertical
lever OB attached at
right angles to the axis
Of a telescope LL, which
is itself pivoted at O and is focussed on a distant vertical scale CC'.
The bar AB is first placed in melting ice and the scale division C
■ en duo ugh the telescope is noted. Next the bar is enclosed in a
hotwater bath. The rod AB expands to B' thereby tilting the
telescope LOL to the position L'OL' and the scale division C is now
seen through the telescope. The expansion BB' is equal to OB tan
L BOB' = OB tan ./ COC = OB >( CC'/QC..
Paschen employed a combination of the micrometer screw and
the lever. The expansion was multiplied in the ratio 1:5 by the
lever and this magnified change in length was measured by the
micrometer screw. An optical lever arrangement is also sometimes
used when the expansion causes a plane mirror to tilt and thereby
deflect a ray reflected from the mirror.
5. Standard Methods. — At the present time the standard methods
employed for measuring expansion are ;
(1) Comparator Method, (2) Henning's Tube Method, (3) Me
thod of Interference Fringes, Methods (1) and (3) are direct while
{2) is indirect.
6. Comparator Method. — This is a standard precision method for
■determining the expansion of materials in the form of a bar or tube.
The bar, about a metre long, is mounted horizontally in a double
walled trough so that it can expand freely at both ends (Fig. 2) and
has two fine marks L, L made near the ends. A standard metre is
i mounted horizontally in another doublewalled trough, both
these troughs being arranged parallel to each other and mounted on
rail: it either "the experimental bar or the standard metre can he
>ughl into the field of view of two vertical microscopes M, M. The
mi< . are provided with an eyepiece micrometer or can be
moved parallel to the direction of expansion by means of a micro
meter screw, and are fixed vertically in rigid horizontal supports pro
jecting out from two massive pillars, the distance between the micro
scopes being about one metre.
160
THERMAL EXPANSION
[CHAP,
First the two troughs are filled with water surrounded by melting
ice in the space between the double walls. The experimental bar
is then wheeled into position and the fine marks on it arc viewed
Fig. 2, — The Comparator method
through the two microscopes and their positions are noted in the
micrometer. The standard metre is then brought below the micro
scopes and the two extreme marks of graduation are viewed through
the microscopes. Front the change in the micrometer reading die
length o£ the bar at 0°C is obtained. The experimental liar is then
heated b\ replacing the melting ice in the double
walled space of the trough by water which is heated
under thermostatic control. The fine marks are
i through the micrometer eyepiece.
Hie increase in length is determined from the
change in micrometer reading. For measure
mem •. temperatures the experimental rod
is placed in a tube which is immersed in a suitable
liquid bath (e.g. liquid air) .
7, Hearing's Tube Method of Measuring
Relative Expansion.— In this method the ex
perimental and the comparison bodies are
together brought to the same temperature and the
differential change of their lengths is measured.
The comparison "body is so chosen that its expan
sion in the temperature region is accurately known
and, if possible, is also very small. Fused silica
serves this purpose well. Inside a long vertical
tube made of some welldefined glass (fused silica,
Jena glass) , there is a ground point, molten and
drawn out of the same glass at. its lower end.
Upon this point rests the experimental ind R
(Fig, 3) , about 50 cm. long and having both of its
end faces ground plane. Upon the upper surface
of R rests a pointed end of another glass rod S
made of the same glass as the outer wider tube.
To the upper end of this rod as well as of the
S — _
worn.
ft'
C ; i
fM t
*«*♦
Fig, 3, — Hetmiog's
apparatus.
outer tube are attached endpieces carrying scales. The whole tube
vn.
FIZEAU S INTERFERENCE METHOD
161
up to half of the height of die rod S is immersed in a hot or cold
bath and the relative shift, of the endpieces is measured with a
microscope provided with a micrometer eyepiece. The shift gives the
relative expansion of the experimental rod against a glass tube of
equal length. This is so on the assumption that the temperature of
the rod and of the outer tube is the same at the same height. For
high and low temperatures suitable baths may be employed.
g. Fizeau's Interference Method. — Fizeau devised an optical
method '.'< .; upon the observation of interference fringes. This
method is capable of very great accuracy and is specially suitable
when small specimens of the experimental substance are available,
as in the case of crystals*
In his original experiment, Fizeau used the substance B (Fig, 4)
in the form of a slab about 1 cm, thick with two of its opposite plane
X'
t
..:
M
Fig. 4. — Fizeau' s interference method,
rail el and polished. It was placed with one of these faces
tl on a metal plate A supported by three levelling screws S, S.
Vh projected upward through the metal plate a little beyond
the upper surface of the slab B. A convex lens L having the Tower
;• large radius of curvature was placed on these screws so
tin i in film of air lay between this surface of the lens and the
upper polished surface of the slab. With the help of a mirror M and
a right ui I d prism P placed above the lens, horizontal rays of light
Erora ;i sodi im fl; ae F were sent down vertically to illuminate the air
film and the i ays reflected repeatedly at the surface of the slab and the
lower surface of the lens proceeded vertically upwards and were again
reflected by the prism P and received by a horizontal telescope T so
that Newton's rings could be seen through the telescope.
11
162
THFRMAL EXPANSION
CHAP.
We know that in the case of Newton's rings the condition lor a
bright ring is
S = 2 t ie cos r « {In f 1) A/2,
where 8 is the path difference, e the corresponding thickness of the air
film and p its refractive index, r the angle of retraction of the ray into
the film, n the order of rings and A the wavelength of the light used.
In the present case ^ = 1, r = 0. We have therefore
8 = 2*= (2n f 1) A/2. .... (4)
The difference in the thickness of the film at two successive bright
rings is A/2, Hence when the thickness of the film changes due to
expansion of the slab and of the three screws supporting the lens the
rings appear to pass across a mark in the lens. Since onetenth of the
distance between successive bright rings could be measured, the change
in length of the order of A/2 i.e. about 0.00002944 mm. could be
determined.
When the above arrangement, producing the air film was enclosed
in a chamber which was heated, the thickness of die film changed due
to the differential expansion of the screws and the substance E, and
the shift oF bright rings across the mark was observed. If x bright
rings are thus shifted, the difference in the expansion of the projecting
portion of the supporting screw and of the slab B along its cnickna
equal to xX/2. In order to find the expansion of the screws* the slab
was removed and interference rings were produced by reflections at the
surface of the lens and the polished surface of the metal plate
through which the screws projected.
►be and Fulfrich improved Fizeau's apparatus by replacing the
screws by quartz rings as shown in Fig. .5, p. 163. G and D are two
quartz plates and R is a hollow cylindrical tripod, also of quartz, cut
with its generating axis parallel to the optic axis, and placed b<
G and D. Tin: specimen is placed inside R and the fringes are formed
by ' nshaped air film enclosed between the lower surface of D
and the upper surface of the specimen, the angle of the wedge being
very small. The light from a Geissler tube (Fig. G) containing mercury
and hydrogen is used. It enters the telescope at right angles, is
deviated through a right angle by means of the prisms P, P and
then falls upon the system as a parallel beam. The fringe systems
for different wavelengths are formed at different heights, in the focal
plane of the objective. By turning a screw any of these systems
ran be brought in the field of view of the micrometer eyepiece. The
lower surface of the upper quartz plate B is provided with a
mark of reference and the number o[ fringes crossing this reference
mark due to rise of temperature can be measured with die help of
the micrometer eyepiece. If X lt A a} A a ,. ...... denote the various wave
lengths of light employed and Xi .£■ § v .\,H s , * a +6, the number
of interference bands displaced across a fixed line (x representing a
whole number and £ a fraction) , the increase A in the thickness of
the ah' film is given by
VII.]
THE FRINGE WIDTH DU.ATOMETRR
168
n : .. rings
of Abbe,
Fig. 6. — Apparatus for measuring expansion
of crystals by Fizeau's method
A= ^(*i+!i)  £ (*+*■)  (*+*>}■
^
9. The Fringe Width Dilatoraoter. — In the last method the change
in length was found from observations on displacement of the fringes.
Priest devised a dilatomctcr in which changes in length can be
obtained from the change in width of the interference fringes. The
aratus is indicated in Fig. 1, p. 164.
The air film is enclosed between the lower surface of the ro
and the upper surface of the base plate, both of which are opti
cally plane and enclose a wedgeshaped space (0.1 to OS mm. thick).
ample under test ends at the top in a fine point X xipon which
rests the cover plate. On looking down in the direction OO, a system
of interference fringes will be seen (as shown in the plan) appearing
to lie in the plane bb so that the fringes and the referent
and xx on die mirror can be simultaneously focussed. When the
sample expands on heating, it tilts the cover plate and thereby
thickness of the air film and consequently the width
of the fringes'. The number of fringes between the lines ss and xx
are observed both initially and finally, and from this the expansion
can be calculated.
1G1
THERMAL EXPANSION
CHAF.
rlr.:;
The calculations can be readily
made. We saw from equation (4) in
last section lliat if the film thick
ness increases by A/2, there is a shift
o£ one fringe across the mark, the
fringes actually contracting. Thus
if in the present arrangement the
number of fringes between the marks
ss and xx changes by x\i), and d
denotes the distance between ss and
and A the wavelength of the light
employed, then the change <t> in the
angle between the planes bb and cc
measured in radians is given by
Again, if D is the perpendicular
distance from X to knifeedge SS, A
the relative expansion of the sample
with respect to a piece of equal height
made from the material composing
the base plate, then ^ is also given by
\— Fringe width ditatCHneter.
6) and (7) we get
**■
£)•
CD
(8)
Knowing A» the coefficient of expansion can be calculated,
10. Discussion of Results*— Table 1 gives die mean coefficient
of. expansion of several substances between and 100 D C. The mean
coel :!■ multiplied by 10 fi is given in the table,
Table 1. Coefficient of Linear Expansion of Substances.
Substance
Ax 10 6
bstance
A >< *0 e
pur °C
per r C
Aluminium
: v.;.:";
Platinum
8,8G
Copper
16.66
Palladium
11.04
Cadmium
SIM
Silver
19.68
Chromium
8.4
'1 ungsten
4.5
Lead
28,0
Brass
18.9
Magnesium
26,0
Invar
0.9
Manganese
22. S
Jena glass
8.08
Molybdenum
5.20
Pvrex glass
3.3
ickel
13.0
Quartz glass
0.510
v.q
EXPANSION OF ANISOTROPIC BODIES
165
But as already mentioned in section 3, these values change
appreciably if the final tempera tore is different from 100 ^C The
mean coefficient A is a function of the temperature. As the final
temperature is lowered the coefficient decreases. Grimeisen lias
found the value of the quantity j ~ for very low temperatures and
has deduced an important law connecting die coefficient of expansion
and the specific heat. Gruneisen's law states that for a metal the
ratio of the coefficient of linear expansion to its specific heat at con
pressure is constant at all temperatures.
U. Surface and Volume Expansion,— The change in area and
volume can be easily calculated from a knowledge of the coefficient
of linear expansion.' A rectangle of sides / and b will, on being
heated, have sides of lengths 1(1 +AQ and & (I + At), and its area
will become lb (1 j Ai) 2 . If the initial and final areas be A and A
we have , A ,
A — A (1 4 2A1) approx., . . . (9)
since A is small. Thus the coefficient of surface expansion is 2 A.
Similarly, the coefficient of volume expansion can be shown to
12. Expansion of Silica Glass, Invar.— Silica glass (quartz which
has been lidified into die noncrystalline form) is now
: i v esipL or the construction ;of thermometers. The
ansion of silica h very small (A = 0.5 X 10 9 per °C.) , and is very
con ntl) determined by Film's method* Vessels made of this
material can be heated without any fear of breaking. The curve
connecting the coefficient of expansion and temperature is a straight
line ihe room temperature and 1000°C. but at both limits
it bends. The coefficient is negative below — 80 C C,
Invar is another special substance, being an a!loy of nickel (fCyf )
and steel. Its coefficient of expansion at ordinary temperatures is ex
tremely small and hence it is generally employed for making secon
dary standards of length, and in the manufacture of precision clocks
and watches.
Anisotropic bodies
13. Tt was first observed by Mitscherlich that the angles be
tween the faces of cleavage^ of a crystal of Iceland spar change when
the crystal is heated. He gave the correct explanation of the pheno
menon, viz., that the expansion of the crystal is different in different
directions and this is the cause of the change in angle. Such sub
ices are called anisotropic or nonisotropic.
For every crystal, however, there can be found three mutually
perpendicular directions such that if a cube is cut out of the crystal
with its sides parallel to these directions and heated, the angles will
remain right ancles though the sides will become unequal. These
directions are called the principal axes of dilatation and the coefficients
of expansion in those directions are called the principal coefficients
166
THERMAL EX P A MSION
[CHAP.
of expansion Denote these by X x , Aj, A^ Then a cube o£ sides U will
on being heated to *°C, become a parallelepiped whose edges will
be given by
, =y*[l + (A* M, +**)']
(11)
The volume coefficient of expansion is thus A, +*«+**■ The Linear ex
pansion in any other direction can be readily calculated in terjns of
the principal coefficients and the direction cosines.*
14. Experimental Methods and Results,— Crystals are best investi
gated by the interference method. The crystal is cut in the manner
desired, into a plate with parallel faces from 1 to 10 mm. thick,
and is placed between the glass plate and the metal disc. The de
link of these experiments have already been given.
When the expansion along the various axes of different crystals
is investigated very interesting results are obtained. In the hexago
nal system, for optically negative crystals the expansion along the
axis is always greater than that along an axis at right angles
to it ; while for optically positive crystals the reverse Is the case.
Thus, for Tcelaud spar we have expansion parallel to the axis and
contraction perpendicular to it. The contraction is always much less
than the expansion so that the volume coefficient remains positive.
EXPANSION OF LIQUIDS
15. In case of liquids we have to consider only the cubical ex
am be expressed as a fu
anperature ; thus
F= P (14«it «,' : — • )»  • (12)
or ap , ,,
r«(j *t), ■ ■ ■ ■ (is)
ere « is called the mean coefficient of expansion between
i t*G. Thus if a mass M of the liquid occupies the volumes F, J ',,
at P and 0°C, the densities p, Po of the liquid at the respective
temperatures arc p =Mf V, p*= M/V . Using (13) we get the relation
" £.l+«fc / . , . (H)
P **
The expansion of liquids is much greater than that of solids, yet
il is more difficult to measure, for it is complicated by the expansion
of the containing vessel. The expansion _ observed is called the
apparent expansion and is a combination of the two effects,
'.. expansion of the liquid and of the containing vessel It can
be shown (see sec. 17) that the coefficient of absolute expansion of
the liquid is approximately equal to the sum of the coefficients of
expansion of the containing vessel and the coefficient of apparent
expansion of the liquid. Thus the former can be determined if
latter two quantities are known,
* Further sec Glazebrook, A Dictionary of Applied Physics, Vol. 1, p. 876.
WEIGHT THERMOMETER METHOD
167
There are three well known methods for determining the appa
rent or relative expansion :— « B jJ,»TW«if!
(i) The Volume Thermometer Method, hi) the Weight Thermo
meter Method,' and (Hi) the Hydrostatic Method.
16. The DSatometer or Volume Thermometer Method The ^
S* divMon of Ltd ! then volumes of the liquid at the two
temperatures are ^ + ^ ^ (F# + ^ (I + ^.
y being the expansion of the containing vessel. The volume at
t°C. is also equal to
(F« + *ift) (! + «*>'
where « is the coefficient of absolute expansion of the liquid, Equa
1 ^ehave^ ^ < 1 + ftsB <r8+% ^ (l + rf>
Knowing y the true coefficient « is calculated, or if 7 is not known,
the relative expansion «■ y can be evaluated .
17. The Weight Thermometer Method.* more accurate method^
deluding upon the determination of weight and not of volume
iskmishld by the pyknometer or the weight ttarai^cki. Thtse
are vessels so constructed as to take a definite volume of t liquid.
The weight thermometer is of the shape shown
in Fig. 8, and is made of glass or fused silica.
It is first weighed and then completely filled with
the liquid by alternate heating and cooling with
the open end dipping in a cup of the liquid.
The experiment consists in weighing the
thermometer filled with the liquid at two tem
peratures. Let it.!, Wi represent the weights
of the liquid filling the thermometer at tern
peratures t x and t 2 respectively. If Vj, F 3 are
the volumes of the vessel at the two tempera
tures \md p v p 8 the corresponding densities el
die liquid, then
w x = F 1PI , u> 2 — Vzp* • (15)
if a, y denote the expansion coefficient
[quid and the vessel respectively,
Fig. 8.— The W i£
Thermometer.
; +vf*
l+rh
w 2 lhyr 2
Pj__l+nh
Pi l+a/2
11 oV
. .  ' . (16)
—X) ih—h) approx. (17)
The apparent expansion a can be obtained from (16) or (17) if
168
THERMAL EXPANSION
GH IP,
the expansion of glass is disregarded i.e., y is put zero. We then
obtain from ;
or
S — 1 } a{t « — U) app rox.
a=* —  L_ approx.
(IB)
Equation (16) can be breated rigorously. Assuming t 2 to re£er
to 0°G, and dropping the suffix I, equation (16) yie
w _ 1 ! ,
BL 1 ! at '
or
a =
wt w
(19)
It is thu^ seen that, rij ly speaking, the true expansion coeffi
cient is a little more than the sum of the apparent coefficient and
the expansion coefficient of glass, though the difference is almost
negligible, and for all practical purposes we can assume a = a$y>
We could treat equation (10) im rally when the result will
be more complicated than (20) . Knowing y the absolute expansion
in be calculated.
The therm ometi nployed to find the cubical expan
ig the specimen inside the thermo
:r.*
18. Hydrostatic Method (Matthiessen's method), — This consists ill
finding the apparent weight of a solid when immersed in the liquid
oectivcly. The loss in weight ol: the
solid is by Archimedes' principle equal to the weight of a volume of
the liq lal to that of the solid; denote this quantity by w. Then
where V Xt V% denote the volumes of the solid at the two temperatures
t lt f 2 respectively. Then
v i _ 1 +y<i Pi_l±al s
K " ! l ;
and proceeding as before,
i ~M? a
P 9 1 4 <*V
a—y =
approx.
(21)
An equation analogous to (19) can also be deduced.
19. Absolute Expansion of Liquids. — As already mentioned, the
three foregoing methods may be employed to find the absolute
expansion of a liquid provided the cubical expansion of the contain
ing vessel [or of the immersed solid in § 18] be known. One way
* Sec Glazebrook, A Dictionary of Applied Physics, Vol, 1, p
vn.
HVOROSTAT1C BALANCE METHOD
169
Balance Method.
of finding the latter is by calculating it : linear expansion. This
is, however, open to objection for Che linear expansion is deter mined
from bars of the material and it cannot be assumed a priori that the
physical i es of the material do not change when it is an
nealed and worked into a vessel of some shape. For this reason it
is best to select vessels of fused silica for which the volume coeffi
cient is extremely small (about 0.0000015 per °C.) .
20. Hydrostatic Balance Method*— There
thod of determining the ahsolute expan
i of a liquid which was first given by
Dulong and Petit. It depends on the hydro
static balancing of two liquid columns at
different temperatures. Dulong and Petit
employed a simple Utube for the purpose.
•Regnault brought the upper ends of the tube
close together, an improvement which made
it easier to observe the difference in height of
the two columns. The diagram (Fig. 9)
scives to illustrate the principle ol the
method. A glass or metal tube, bent as
shown in the figure, contains mercury. The
vertical columns AB, CD, C'D r are sur
rounded by melting ice and are thereby
maintained at (PC, while the column A'B' is surrounded by an oil
maintained at any temperature i°G. Suppose that AA' is
zontaL Let H, H r t h, h* denote the heights of mercury in the various
columns as shown and p, p Q the densities of mercury at t a C, and 0°C.
Then since the pressures at D and D' are equal, we have by equating
the two expressions Cor the hydrostatic pressure at A,
h?p t +H , t> = Hpo+hfr. .... (22)
But
where c is the coefficient of absolute expansion of mercury. Hence
j^+AWf+A, .... (23)
whence c can be calculated.
If the columns H, h, h f are not at 0°C. but at temperatures
, f 3 respectively we shall gee
*: + . A ' . _ .jl h t . . ( ,
1+rf ^ l+c^ ' " 1 + ^f, r Hc a '« '
where the quantities c,, c>, c ? , denote the mean coefficients of expan
sion between the different ranges. These can be determined by
having the temperature of A'B' to be £ 1? t%, ( s successively,
the height of A' above A is /i la a corresponding term can be adr
the righthand side.
Regnault's observations, though carried out with greal
must be corrected for various sources of error and hence cannot vi: Id
170
1H BRM A I, EXPANSION
CHAP
n
rttiti eta ij
CH
Fm. 10.— Arrangement of
Callendar and Moss's
apparatus.
results of high accuracy, Callendar and Moss repeated the experi
ments aiming at a high degree of accuracy. Instead of a single
of hot and cold columns 1.5 m. long employed by Regnault, they
wsed six pairs of hot and cold columns each 2 m. long and connected
in series as shown diagrammatically in
Fig. 10, The hot and cold columns are
marked H and C respectively. The differ
ence in height of the first and the last
column (viz., &b) is six times that due to
a single pair.
In the actual apparatus ef, gk , . . were
doubled back so that all the columns marked
C were one behind the other, and similar
was die case with H columns. All the H
columns were placed in one limb of a rect
angle and all the C columns in one limb of
another rectangle, while the o titer limbs of
these rectangles contained electrically heated oil and icecooled baths
respectively. These were kept circulating by means of an electric
motor and their temperatures were determined by a long 'bulb' re
tan cc LliiiriTometer, Experiments were performed in the range to
300°G. and an accuracy of 1 in 10,000 was aimed at,
21. Results for Mercury.— Their values for mercury are, howe
ifEerent from the mean of earlier investigators such a*
; . Harlow hi made accurate dctermi
i the help of a weight thermometer of silica and aimed at
of 1 in 18,000. The concordance o£ results with bulbs of
,ed that as quite isotropic. Th ■ ient
of e i in the region to 100°C. is according to Callendar and
C.
The following numerical examples show how the expansion of
rcury Is taken into account in (a) the correction of the barometric
,ture, (b) the correction for the emergent colum
of a mercury thermometer and (c) the compensation of the mei m
ttm.
Example 1 . — A barometer having a steel scale reads 750.0 mm. on
a day when the temperature is 20°C. If the scale is correctly 'gradual
at o G., find die true pressure, given that the coefficient of linear ex
pansion of steel = 12 X 10~* C C~\ and coefficient of expansion (abso
lute) of mercury = 182X IO  ' 5 per °G
The length of scale at 20"C = 75.00 (I + 20 X 1£ X MH") cm.
Density of mercury at 20°C ^= p f (1 + 20 X 1S2 X 1{) ~ 6 )
where p Es density of mercury at D C.
75.00(1 4 20 k 12 x 10^)^
ressurc —
dyncs/cm a *
(1+20*182x10^
=75.00(10.0034) ^=74.745 Ptt g dyncs/cm.*
VII.]
EXPANSION OF WATER
171
Example 2. — A mercury thermometer, immersed up to S0°C mi
in a hot liquid reads 230*0. If the exposed stem has an a
temperature of 50°C, calculate the true temperature, given that
mean coefficient of expansion of mercury is 182 >< 10* per n C, and
coefficient of linear expansion of glass = 8 X W 6 D C 1 
Coefficient of apparent expansion of mercury
= (182  3 X 8)10* = 158 X 1CM °C^
Hence, the exposed stem, if at the true temperature t*fcl., would
occupy a length (23030) [1 + (£50) 158 X 10"*].
/. (  230 = (230  50) (i  50) 158 X H)*,
when * = 235 U C.
Example 3. — F In a mercury pendulum a steel rod of length I cm.
D C supports a glass cistern containing mercury, find the height
to which mercury should be filled up in the cistern for perfect compen
sation of the pendulum, given that the linear coefficient of expansion
of steel a '= 12 X 10 G , linear coefficient of expansion of glass g = 8,5
X 10"*, cubical coefficient of expansion of mercury m = 1.82 >( 10'
I .et h be the required height of mercury in the cistern at C
V the volume of that men nd A the crosssectional area of the
at n C. Due to the rise of temperature to t c 'C, the
volume ' ' 7 (1 J mt) , the crosssectional area of
the cistern increases to A (1 2gf) and therefore the height of mer
cury increases to
Since the centre of gi \\ ic of mercury at 0°C is at. a height ft/2
from the bottom, it will rise to (ft/2)(l \mt2gt) at t a C, the in
crease being (h/2) (mt — 2gl). The increase in the length of the steel
red at t Q C is kit. For perfect compensation these two changes must
be equal. Hence
imt2gt) = lot.
Oi
h =
2 a
22
•';
= _2xl2xl0^_
(18217) xlCr* ' tM * J <"
22. Expansion of Wafe*. — Tt is wellknown that the expansion
of water is anomalous in the region to 4°C, Several workers such as
Hope, Despretz, Ma tth lessen, Joule and Play fair and others, measured
this expansion carefully. A constant volume dilatometer (sec. 16)
may be employed for this purpose. If the dilatometer is made of
ordinary glass some mercury is initially put in it to compensate for the
expansion of glass. Since the expansion of mercury is 0.000182 and
of glass 0.0000255, a volume of mercury equal to dueseventh of the
volume of the dilatometer will be required for compensation. These
experiments show that when water at 0°C is heated it goes on con
tracting as long as the temperature is below 4°C. Above 4°(] it
172
THERMAL EXPANSION
[CHAP.
expands on heating. Accurate experiments by Joule and Playfair
show that this temperature of maximum density is 3,95 °C. This
anomalous behaviour is usually explained on the assumption that, there
exist three types of molecules H 2 0, (H 2 0) 2 , (HgO) 8 , which have
different specific volumes and are mixed in different proportions at
different temperatures. The total volume occupied is assumed to be
the sum of the specific volumes, though there seems to be little
justification for such an assumption.
PRACTICAL APPLICATIONS OF EXPANSION
23. The expansion of solids and liquids is of great importance in
daily life and its consequences have often to be borne in mind
carefully. The student will be familiar with most of these from his
elementary studies. Thus it is wellknown that allowance must be
made for expansion in the laying of railway lines and the erecting of
.steel bridges, the contraction of metal tyres on cartwheels etc. The
expansion of tine steel scale must be taken into account in reading
the eter (sec. 2.U Example 1). Similarly the expansion oi
mercury must be found out for obtaining the true temperature fro
a mercury thermometer (sec. 21, Example 2). Expansion in optical
and electrical apparatus also causes difl i. The mirrors of
reflecting telesc ich are very accurately figured must be pro
tected from distortion of the ce by expansion of dij . por
different temperatures. The i tee of a
coi; on its diirif.r changes with expan
sion con.
In dealing with glassware it is particularly Important
•i and contraction as the glass is lik
to ;:• I to sudden changes of temperature. Thii
oor condi heal and the difference in tem
iure between the different parts causes unequal expansion.
;e is avoided by either choosing thin glassware or choosing
: ; of small coefficient of expansion and high thermal con I [vity.
Fused si sels ran he heated to white heat and plunged safely
into cold water, while pyrex glass can be thrust into a blowpipe flame
suddenly without risk of fracture. Platinum wires can be sealed into
ordinary glass (lead glass) without any risk of a crack developing
because platinum and glass have practically the same coefficient of
expansion but a copper wire would cause cracks to develop due to un
equal expansion.
24, Compensation of Clocks and Watches. — For our present pur
pose the pendulum of a clock may be treated as a "simp i . du
rum" with a bob of negligible dimensions suspended at the end of a
wire o£ negligible mass. "The time kept hv the clock depends upon the
time of oscillation of its pendulum which, in the case of a simple
pendulum, varies as the square root of the length or the pendulum.
The length of the simple pendulum is the distance between the point
support and the centre of gravity of the bob. Thus if this length
COMPENSATION OF CLOCKS AND WATCHES
173
vn.j
increases in summer due to expansion, the time o£ oscillation will
increase and the clock will lose time. If the length decreases due to
fall of temperature, the pendulum swings faster and the clock gains.
Therefore unless the pendulum is compensated against the effect of
expansion the clock will gain in winter and lose in summer. This
compensation is brought about by the use of two expansible materials
so arranged that the expansion of one is compensated by the expan
sion of the other.
In the gridiron pendulum alternate rods of steel and brass are
connected as shown h\ Fig, 11 so that the steel
rods expand only downwards, and the brass rods
only upwards. It is so arranged that 7,cu=ta*2
where l x and l 2 are the total lengths of the steel and
brass rods respectively, and «i, og the respective co*
efficients of linear expansion. Under these condi
tions the effective length of the pendulum remains
constant at all temperatures and the pendulum is
compensated. It will be seen that J a /1 2 == W a i ~
3/2 a pproxi m ate ly,
In the mercury pendulum a long steel rod
carries, in place o the ordinary bob, a frame fitted
with two glass cylinders containing ary. Com
pensation is obtained by the expansion of the rod
downwards and the expansion of mercury upwards,
(See sec. 21, Example o) .
Fig, II, — Gridiron
pendulum
In watches the rate of movement is governed by
the oscillation of a small flywheel called "the balance
wheel which is itself controlled by a hairspring. The
time of oscillation of the balance wheel depends
upon the stiffness of the hairspring and the
moment of inertia oE the wheel about its axis of
oscillation. The moment of inertia depends on the diameter of the
wheel and is mainly contributed by the three small weights (Fig. 12)
screwed on the rim of the wheel. A rise in tempera
ture weakens the hairspring and increases the dia
meter of the balance wheel. Both these changes
cause the oscillations to become slower, the former
effect being more important. To effect com]
tion, the rim of the balance wheel is made in
segments of a bimetallic strip with brass on the out
side and steel on the inside, and weighting the rim
with three weights as shown in the figure. Due to
the larger expansion of brass the end of each segment
curls inwards when the temperature rises thereby
reducing the moment of inertia sufficiently to compensate both for
radial expansion and weakening of the hairspring. The precision
watches such as chronometers are usually provided with cor i
balance wheels. Invar steel is now frequently used in the manufacture
Fig. 12.
Compensated
balancewheel.
174
THOtMAL EXPANSION
[CHAP.
of pendulums and balance wheels on account of its very small coeffi
cient of expansion,.
25. Thermostats. — The property of expansion is often utilised for
*
tnr
13.— Toluene Thermostat
constructing thermostats. In these the
temperature of any substance can be
kepi constant for a long time. For
temperatures upto lGG a C s a toluene
thermostat may be used but 'for tempera
tures above JOCC, a bimetallic thermo
regulator is generally used. These will,
be described. '
Toluene Thermostat, — A toluene
thermostat is shown in Fig. 13. In the
bulbs T there is toluene, alcohol or
some other liquid having a large co
efficient of expansion. These bulbs
are immersed in the bath whose
temperature is required to be maintained constant. In case the tem
perature of the baLh increases the toluene expands and forces the
mercury (shown black in the figure) up the tube in C and thereby
ises the opening leading from A to B. The bath is heated by a
burner and the gas ' his hunter is supplied through A and this
opening. Thus, on account of the expansion the gas supply is
1 the temperature falls. Tf the temperature of the bath
falls too much, toluene contracts, the opening is increased and
By adjusting the amount of mercury in K. the
ire can be kept constant at any desired value. In case of
trical heating electrical contact can be arranged above
the mercury lev:: ft.chmg off the heating current.
illic ThermaRegulator. — The Dekhotinsky bimetallic thcr
ulator is shown in Fig. 14. It. consists of a compound strip R.
which is made by welding to
gether brass and invar steel.
The scrip is wound into a small
helix and is connected by
means of the rod P to the con
tact maker C which makes or
breaks the electrical circuit of
the heater H. When the tem
perature rises the expansion of
the brass causes the helix to
unwind and thereby the con MAINS
tact at C is opened thus break „...„._,,,,.„. ^ _ . .
. . £ , • ., ,,,., Fig. 14. — I be Bimetallic ThermoRcgulator.
:c electrical circuit. With
c the helix contracts and thus contact is made at C
thereby completing the electrical circuit. With this apparatus tem
peratures up to 3(K)°C can be maintained to within ±1*C. for a
long time.
VII. I
EXPANSION OF GASES
175
(27)
EXPANSION OF GASES
26* Expansion of Gases*— The expansion of gases forms the very
basis of the system of thermometry and the perfect gas scale discussed
in Chap. I. As stated there the results are best expressed in the form
of Charles' law which holds very approximately for the socalled
permanent gases in nature. Here we shall describe the experimental
methods of determining the coefficient o£ expansion.
In the case of gases it is necessary to distinguish be L ween
coefficients of expansion : (1) the volume coefficient of expansion « at
constant, pressure, and (2) the pressure coefficient of expansion $ ai
constant volume. The volume coefficient of expansion is denned
as the increase in volume of unit volume at a C for each centigrade
: ■■:.?. rise of temperature at constant pressure. Thus
<^ = K V J L . (25)
where V and F n denote the volumes of a fixed mass of gas at t° and
II : C, Or
F=F*tl + «9 (26)
If the volumes at temperatures t\ and f 2 a* 6 Vi and F 2 respectively*
get, with the help of (26).
«~ Eta ....
We can thus determine a by measuring the volume of a fixed mass
of gas at two temperatures.
Similarly the pressure coefficient of expansion of a gas is defined
as the increase in pressure, expressed as a fraction of the pressure
at 0°C. for one centigrade degree rise of temperature when a fb
mass of the gas is heated at constant volume. Thus if p and p^ be
the pressures at t Q and 0*C we have
from which relations analogous to (26) and (27) can be deduced.
27. Experimental determination of the Volume Coefficient of Expan
sion. — GayLussac was among the earliest to measure the volume
'licient accurately. Resmault used an improved form of apparatus
and corrected his results for various sources of error. Fig. 11 shows
a laboratory arrangement for determining the volume coefficient and
employs Regnault's technique in a simplified form.
e bulb A is connected by a narrow glass tube to a calibrated
limb B of a mercury manometer whose other limb C can be moved
up and down for adjusth ry level in. B. The tap T enables
the quantity of gas in A to be adjusted. First the bulb A is put in
j cold water bath at t t °, and after it has acquired the temperature
of the bath, the tube C is adjusted until the mercury stands at the
same level in both arms, and the mercury level in B noted. Then the
*
176
THERMAL EXPANSION
[CHAP*
is heated, when the enclosed air expands pushing the mercury
down in B and up in C, The bath is maintained at a certain tempe
rature and the tube C lowered to bring
the mercury level at the same height
in B and C, and Uie volume of gas in
B read from the graduations. The
process is repeated for every 20° rise
of temperature up to I00 c C, and the
observed readings are utilised for cal
culating a from (27) .
The results are hest treated by
plotting the observed volumes against
temperature on a graph* It is found
that all the points lie on a straight line
showing that equal changes in tem
perature lead to equal changes in
volume at constant pressure. This is
Charles' law which may be formally
stated : for a fixed mass of gas heated
at constant pressure, the volume in
creases by a constant fraction of the
■me at 0°C for each cem .
ree rise in temperature. This also
follows from the result that a comes
out to be the same whatever values of
2 are utilised in (27), Results
further show that a = 1/273 (nearly!
...'..
Hed permanent gases
For accurate work ecu
)e applied for the following
sou, : — (1) the gas in the
ow tube and the manometer is at
a different i emperature From the bath,
on of the glass bulb
with rise of tern re. Regnault
appli erections for these and
found that all real gases showed small
departures from uniform expansion
and that the coefficient of expansion
red slightly from one gas to
another.
28. Experimental determination
of the Pressure Coefficient of Expan
sion.— The pressure coefficient can be
y determined in the laboratory
with tiie help of an apparatus
known as Tory's apparatus. It
consists of a glass bulb A, of about
1 C.C. capacity, which is filled with
dry air and is connected by a glass
Fig\ 16. — Joly's apparatus
for determining pressure coefficient.
VIL]
PRESSURE COEFFICIENT OF EXPANSION
177
capillary tube to a mercury manometer mounted on a stand (Fig, 16) .
A fixed' reference mark X Ls made on the tube B near the top by
means of a file, and the mercury level is always brought to this mark
by adjusting C before any reading is taken. This ensures that the
volume of the enclosed gas is kept constant. A metre scale S is fixed
to the vertical stand to read the difference ft in the levels of mercury
in the two tubes B and C. The bulb A is immersed in a water bath
which is well stirred, and temp.Liai.Lin: are read with a mercury
thermometer.
First the experiment is done with cold water in the bath and
die difference h m the levels of mercury in the two columns noted.
If the barometric height is H, the pressure of the gas is H ±h depend
; ion whether the level in C is higher or lower than that in B.
The bath is dien heated through about 20°, heating stopped, and
the bath well stirred. The mercury is again brought to the reference
mark and the level watched carefully. When the level becomes steady
at die reference mark, the reading in C is noted. Heating is then
resumed and readings are taken in this way at intervals of 2Q\
The
j>
(29)
coefficient of expansion /3 is then calculated from the relation
Pz Pi
P±h—p.: :
For accurate work various corrections are necessary. The most
difficult to estimate is the "dead space" correction (p. 9) since the
exact temperature of the gas in the capillary tube is not known.
The expansion of the bulb introduces an error in /S of the order of
1% ; this can be satisfactorily corrected for by adding the coefficient oi
cubical expansion of glass to the observed value of p. Experiments
shown that p is fairly dose to 1/273 for all the permanent
.•.•; which means that for a fixed mass of any gas heated at constant
volume, the pressure increases by 1/273 of the pressure at 0°C for
each centigrade degree rise in temperature. Accurate experiments
however show thai "this expansion is neither uniform for a gas, nor
is it exactly the same for all gases.
As mentioned in §7, Chap. L these observed deviations of a and j3
from the correct value of 1/273.16 are really due to the deviation of
actual gases from Boyle's law. Let a fixed mass of perfect
which by definition obeys Boyle's law, have the pressure ad
ioe t'o at 0°C, When it is heated to t°C, the product of pre
and volume can be written as jfy>#o (1 + at) or ptft/tt (1 f /?r) depending
on whether the pressure is kept constant or the volume is kept cons
Since these products must be equal by Boyle's law, we get a — /3
lor a perfect gas. The expansion coefficient is experimentally found
to be 0.0036608 for all gases provided they are reduced to the state of
a perfect gas (p— >Q) . "The equality of « for all gases is really a
consequence of the kinetic theory.
Book Recommended
A Dictionary of Applied Physics, Vol 1,.
pp.
1. Glazcbrook
872—900.
12
CHAPTER VII!
CONDUCTION OF HEAT
1. Methods of Kent Propagation. — When a bar of metal is heated
at one end and held at the other by the hand we ■ nsation of
heat Heat travels from the hot end along the bur and produces this
sensation in the hand. The power of transmitth manner
is possessed by all substances to a varying degree and this phenomenon
tiled Conduction oj Heat, In the process of Co m, heat in
transferred by the actual motion of heated parricli s of matter whether
liquid or gaseous. Tins is best illustrated by placing gently some
ils of potassium permanganate ;il the bottom of a beaker conn
f ainhig water and heating it. Heated water rises up and curls down
forming a closed path which is rendered visible by the red colour
imparted to the water. In conduction, heat is transferred by 'contact*
and there is no apparent transfer or matter.
In both the above processes the intervening medium takes an
active part in heat propagation, I mi in addition to these there is
anoiJi: i i . ii ■■ the intervening medium takes no part, il
id near a coal furnace, we feel the sensation of heat. If we
■ the iuzi we fee] w\ are the source of heat is the
e sun, and In the latter case it is at an enormous distance
io mate] ial medium b< tween us am
M.. This phenomenon
is call r'hc processes of conduction and convection arc
./., due to the action of the intervening medium while
radiatii with the enormous velocit; Eit. In this chaptei
aduction of heat.
2. Conductivity of Different Kinds of Matter. — Common observa
show that different substances vary enormously in their con
ducting power. A glass rod can be melted in a flame by holding ii at
[nt two or three inches away from the Game, while a copper rod
under similar conditions becomes too hot to Couch. Copper is thus
i better conductor of heat than glass. Metals, in general, are good
conductor, ol heal ; glaj iod and other nonmetals are bad con
ductor*, Hold inside a flame two blocks, one of wood and the othev
of copper, each covered with paper. The paper covering the copper
1 I'M i: i , not burnt, for heat is rapidly conducted away h r and
mperature does not rise it .nition point. The' paper cover
he wood is burnt, Similarly, wat< be tied in a cup of
paper for the; heat, is taken up by water (convection) ,
Liquids, in general, are worse conductors of heat than solids and
are the m \ very simple experiment shows thai water is a
bad conductor of heat. Take some water in a testtube and sink
into it a piece of ice by weighting ii. The water at the top can be
boiled by Ilea ting it locally while the ice at the bottom docs not even
VIII.]
DEFINITION OF CONDUCTIVITY
179
melt. Air and gases in general, are even worse conductors than water.
Woolen clothing protects us from cold on account
of the fact that it contains air in the interstices which
renders ir a very bad conductor of heat,
The good conductivity of metals is utilised in the
construction of the Davy safety lamp. The flame is
enclosed in an iron gauze chamber (Fig, 1) and can
be taken down a mine. Any combustible gas, if
nes in contact with the naked flame and
burns inSide the chamber. But the iron gauze con
ducts awa; heat so quickly that the temperature at
any point of it does not rise Co the ignition point and
the outer gas does not ignite .
3. Definition of Conductivity.— The first, to give
a precise definition of conductivity was Fourier who
in his memorable Theorie Analytique de la ChaJeur
(1822) treated the subject of heat conduction in a
masterly way and placed it on firm mathematical
basis. We shall first discuss the (low of heal inside a
bar heated at one end. Consider a thin wall of the material with
u irallel Fa< i . such that heat Hows in a direction perpendicular to the
Lftl Ou 0fl be the temperatures of the two faces, I the thickness
rea of each lace, then it can be shown experi
mentally that the amount of heat Q flowing through the wall in a
time f, when the temperature at every point of tHe bar is steady,
is directly proportional to (t) Qi6 2 > 00 tn e area A of the surface,
{Hi) the time t and (iv) inversely proportional to I, that is.
Fig; 1, — Davy
Safety Lamp.
Q,= KA
l
(1)
The coefficient K is a quantity depending upon the nature of 1 1 1 ;
iubstance and is called its thermal conductivity. From this relation
conductivity may be dqfined as the quantity of heat flowing per
tnd through a unit area of plate of unit thickness, when the
difference of temperature between the faces is unity. In analogy
with the electrical resistance, the inverse of K may be called the
distance of a unit cube,
\Yi,v imagine the thickness of the plate to be diminished indefi
nitely. The limitino Milne of * , a or , l is — and denotes
/ / dx
the temperature gradient at any point. The minus sign has been
used befi n ", because the symbol d always stands for the increase.
Hence, the quantity of heat flowing in the positive direction of x
in time df across the isothermal surface of area A at any point x is
given by
dKA^di.
(2)
180
CONDUCTION OF HEAT
CHAP.
This equation is of fundamental importance in the theory of heat
conduction. The Links in which conductivity is measured in the CCS.
system are the calorie per second per square centimetre of area lor a
temperature gradient of 1 G C. per cm. [cal cm, 1 seer* 1 ''C 1 ]*
CONDUCTIVITY OF METALS
4. We shall now give the various methods of deter mining the
thermal conductivity of metak. Methods I to III employ stationary
heatflow while in IV a stationary periodic flow of heat is used.
I. Conductivity from Calorimetric Measurement
5. The definition of conductivity from equation (1) provides a
simple method of determining the conductivity of a substance. We
need have only a slab of the material of known crosssection heated
at one end, and measure the amount of heat that flows out at the
other end in a known time, as well as the temperature of the two
faces. Titus all the other quantities except K in equation (1) are
known, and hence K can be evaluated.
The method though simple presents considerable experlmem
difficulties. It is difficult to measure accurately the temperature of
the two faces of a slab of metal. This is best achieved by keeping
.embedded in the surface the junction oF a thermocouple, the use of
mercury thermometer or resistance thermometer being inconvenient
or impossible. Some early experimenters used steam to heat the slab
at one end and ice or n cool it at tfoe other end, and assumed
ture of the end rare, of the slab was that of steam
and water res But if proper precautions be not taken, the
od .sometimes gives absurd results This is on account of the
fact that a thin film of fluid always at rest, is formed in contact with
the si and this has a large temperature gradient. Hence it is
essential that we should observe the temperature inside the slab itself
by means of thermometers. There are many apparatus based on this
method, and one due to Searle is described below.
Fig. 2 shows the apparatus diagrammatically. One end B of
the rod AB i& enclosed in the steamchamber S while the other end
A projects into another chamber C through which cold water circulates
as indicated, the temperatures at entry and exit being T a and T 4 res
pectively. Temperatures at two points along the bar are measured
by thermometers T x and T £ ; let these be T x and 7V The whole rod
is wrapped round with some nonconducting material Tike worn 1
etc. In the steady state if m grams of water How past the con;
of the bar per second, the heat conducted by the bar per second is
T —T
1 r— S , where d is the distance
?n(Ti—T s ) and this equals &K
thermometers T lt T
Thus, the conductivity K is
between tin:
determined,
In the foregoing experiment some heat was lost bv radiation
from the sides of the bar. This is a source of error and is very
vni.
CONDUCTIVITY FROM CALORIMETRIC MEASUREMENT
181
easily eliminated if the bar be surrounded with some material at
the same temperature as the adjacent portions of the bar. There
will be no How of heat perpendicular to the length of the bar as no
Fig. 2. — Searle's appara
temperature gradient exists in that direction. The surrounding
material is called the 'guardring'.
Beige l utilised this guardring method for determining the
conductivity of various substances such as copper, iron., brass and
mercury. A vertical cylindrical column of mercury is surrounded
by an annular ring of mercury. In both the upper surface of mercury
is heated by steam while the lower surface rests on a metal plate
Thus the temperature gradient as well as the
temperatures at the same level in the experimental column and the
'surrounding annular ring are identical! Under these conditions
there can be no lateral flow of heat and the annular ring serves as
a 'guardring'. The lower end of the experimental column projects
into die bulb of a Bunsen icecalorimeter and the heat flowing out
at die lower end is found from the indications of the mercury
thread. The difference of temperature between several points along
the column is determined by four differential thermocouples. The
conductivity K can be calculated from formula (1) .
This method has been adopted by a number of workers, notably
by Lees, Donaldson, Honda and Simidu, and others. Lees used
the rod method for measuring the con
ductivity of many metals throughout the Kf
temperature range 18°C to  170°C The
imental rod, about 7 cm. long and
0.58 cm. in diameter, was heated electri
cally at the upper end, its lower end being J£
fixed to the base of a hollow, closed,
cylindrical shell, of copper which complete
ly surrounded the rod. The outer cylinder
was suspended inside a Dewar flask and immersed in liquid air or
heated electrically and thus the desired temperature of experiment
i jl
p.
Fig. 3. — The twoplate *
system.
182
CONDUCTION OF HF.AT
CI3AP.
was attained. Knowing the electrical energy spent in the heater
wire and the temperature at two points on the rod by platinum
resistance thermometers, the thermal conductivity can be calculated.
Some corrections are however necessary which are difficult to evaluate
accurately.
Honda and Simidu employed the two plate method shown in
Fig'. 3. P, P 2 are two exactly similar plates made of the experimen
tal material. Between these is symmetrically placed the electrical
heater H in which the heat Q is generated. ' For perfect symmetry
two cold plates, K, K maintained at the same temperature are placed
at the other ends of these plates. Thus the amount of heat flowing
through each plate is Q/2,
IL Conductivity from Temperature Measurement —
Indirect method
6. Rectilinear Flow of Heal, Mathematical Investigation,—
Consider a metal rod heated at. one end.
the isothermal surfaces being parallel
planes perpendicular to the length of
tlie rod, and let the axis o[ x be normal
to ihe.se planes. At a distance x from
the hot end (Fig. I) let Q be the tempera
ture and — the temperature gradient,
t of thickness v at this point. The amount of
di)
(from equa
tion 2), Qz the heat which leaves the layer at the face x\dx is
1
6+dd
Qt—
Lq '
Heale
r o~ 
r
X
3 a
k— Flow of lieat in a rod.
heat Q, whid . the lavei per second is K.
dx
—KA
ii'
since $ ; is the temperature of that face.
dx
Hence the gain
of heat by the layer is equal to
Now before the steady state is reached this amount of heat raises
the temperature of the layer of thickness dx. Let p be die density
and c tiie specific heat of the material per unit mass, and let the
rate of rise of temperature be denoted by  . The mass of the layer
Idx. Hence, neglecting the heat lost by radiation from the
surface, we get
KA—ra dx = pAdx. C
di
JC dW_ _ d*9
pc dx* " dx*
 (6)
via.]
rectilinear elow of heat
183
where h = K/pc Thus h is equal to the thermal conductivity divided
by the thermal capacity per unit volume. This constant k has been
called thermal difjttsivity by Kelvin and tkermonwtric conductivity by
Maxwell but the former term is more commonly used. It will be
seen that the thermal dilfusivity h represents the change of tem
perature produced in unit volume of the substance by the quantity of
would flow in unit time through unit area under unit
temperature gradient. Thus for calculations of the rate of rise of
temperature the constant k is of greater importance than K as equa
tion (0) shows (as the rate depends not only on K but also on pe) t
but in the steady state the rate of flow of heat depends only on K
and not on p c (see equation 8).
Tf, in addition, the sides of the bar are allowed to lose heat
by radiation, this must be taken into account. If E is the emissive
power of the surface, p its perimeter and $ the excess* of the tempe
rature of the surface over that of the surroundings, this radiation
loss 3 , assuming Newton's Law of Cooling to hold true, is equal
to Epfi dx. Hence we should rewrite (5) as
d&
Or
where
KAi.dx^p Adx.c Tl
ax at
m
E P
pAc'
'litis is the standard Fourier equation for one dimensional
flow of heat and any problem in thermal conduction along a rod
consist: simply in the solution of this differential equation.
Steady State — A state is said to be steady when the tempera
ture at every point of the rod is stationary, Le. 3 j  — 0. We have
then
where
ST 5 *  ««l
m' = !i = Kp .
h AK
(*)
If the radiation losses from the sides can be neglected, this
equation further reduces to
dx*
The solution of this differential equation yields
= Ax j B,
♦ When the radiation toss term is included as in equation (7), 6 must be
nuflsared as excess over ike: temperature of the surrouudinjrK. If this term is
not included as in equation (6), S may denote the actual temperature or the
excess over the surroundings.
CONDUCTION OF HEAT
CHAP
where the constants A and B can be determined from the boundary
mnditions. Let these conditions be (1) $ —8 at x = 0, O being
the temperature o£ the source ; (2) 9 = 0± at x = I. Then we have
,'
■x,
. 
where is the temperature at any point x.
If radiation losses are not negligible, the solution will be different,
Let us assume that in this case & = e" is the solution. Hence, by
substitution,
n 2 — w 2 , or n = ± m.
Therefore, the complete solution is
= Ae™ f Be~™ . , . . (9)
where A 3 B are constants,
7. Inge»Haas56 f s Experiment*— A method of comparing the con
ductivity of different substances based on this solution was employed
by Ing» •:' • early as 1789 and is generally shown as a
' l experiment. Bars of different substance?; are coaled
with wax and have their one end
immersed m a hot bath of oil (Fig. 5) .
The wax nielts to different lengths along
different bars. Before the steady si
is reached the U tire at any point
depends on h, i.e., on both the thermal
capacity and the thermal conductivity as
in;! nation (7). This is why
initially the temperature wave is
i . red to travel faster along bismuth
ii along copper, for the low thermal
capacity of the former more than com
pensates for the larger conductivity of
latter. When iJi state is attained, however, the wax is
It on the copper over a greater length,
Let l lf l 2 . . . . . denote the lengths along which the wax has
melted on the differer tfc siperature of the bath, measured
above that of the surroundings, and 8i, the' temperature of melting
wax similarly me;. i the bars are long enough the temperature
at their other ends is the same as that of the surroundings, i.e.,
= 0. The complete solution of this problem is represented by
•lion (9). The boundary conditions for all the bars are (i) 0==O
at *= co, (it) #=0o at ,v— 0, (tit) 0=0 t &tx=L
By substitution in equation (9) condition (i) gives A = G. Con
';, then Lives R=0 O . Tlie solution then becomes
Condition (Hi) then gives
tfiflfr' (10)
or ml = iog,T— .
tus of
Ingenlj
*
V)il.] FORBES* METHOD
Since log,(V*i) is tfte same £or a]1 the bars we haVC
mik = mJj — rnJz =s * • • = constant,
which from the definition of m implies that
/ s — J a
= s n — \ — . . . . = constant,
(ID
provided the different bars have the same crosssection, perimeter
and coefficient of emission. Thus
K
K m
i _
(12)
Therefore the conductivities are in the simple ratio of the squares
of the lengths along which the wax has melted, and if the conducti
vity of one of the bars be known the conductivity of the others can
be 'calculated. This is an indirect method. In order to secure the
same coefficient of emission the bars are electroplated and polished,
& Experiments of Despretz, Wiedemann and Franz. — Desprctz, as
early as 1822, compared the conductivity of two substances by n
of three temperatures at equal aces. The bars were heated at
one end and were provided with a number of equidistant holes
ii ( ughout their entire length. These holes contained mercury in
which" the bulbs of mercury thermometers were immersed for record
ing the temperatare. The theory of the method can be worked out
b the help of equation (9).
Wiedemann and Fran/, following the same principle devised
a more accurate apparatus. The bars under test were about half a
metre long and mi . En diameter and were electroplated One end
of the hair was heated by steam and the remainder surrounded by a
constanttemperature jacket. The temperatures at equidistant points
were measured by a sliding thermocouple which could be mani
pulated from outside.
Fig. 6. — Forbes' apparatus, Statical exper' '
(The dpttcd curve shows fall of temperature along the bar and tangent
to the curve jives the temperature gradient).
186
SUCTION OF HEAT
CHAP
ill. Conductivity by a Combination of the Steady
and Variable Heat Flow
9, Forbes' Method. — One of die earliest, methods of determining
the absolute conductivity of a substance is that due to Fo bes,
Though simple in principle., it is exceedingly tedious in practice, Forbes
used a bar of wrought iron 8 ft long and \\ inch square section. One
end of the bar (Fig. 6) was heated by being fixed into an iron crucible
containing' molten lead or silver, A number of thermometers, with
their bulbs immersed in holes drilled into the liar, were employed to
indicate its temperature throughout its entire length. After about
six hours the temperatures at all points become steady and are read on
the thermometers. The temperature distribution is indicated by the
dotted line in Fig. 6, and follows the law =&&""* *. T h 'S is called
Lhc statical experiment since it deals with the steady state of heat
flow.
To obtain the heat flowing across a particular crosssection,
Forbes determined the amount of heat lost by radiation by the
portion of the bar lying between that crosssection and the cold end.
These two quantities are obviously equal in the stead since
ho heat, flows out from the cold end, this being at the room
ture. Forbes achieved it by performing the dynamical experiment*
so called because the temperature in this case is changing. For this.
purpose a bar only 20 inches long but in all respects exactly similar
to th< y'tv was used. First a high uniform teraperati
iiinicated to this bar which is then allowed to cooi iii. exact!)
the statical bar, and a cooling curve plotted
for it (Fig. 7).
; calculate the amount of heat lost by the statical
bar from the point
x = x x , to the end of
the bar (x = I) . The
amount of heat lost
per second by radiation
from the surface of the
bar from x to x + dx in
the steady state, is
d&
= v
}<i
,U>:, P c w ,
Fig, 7, — Temperature curves in Forbes' bar,
where p is the density of the material, c its specific heat and j
the rate of cooling of that element. Hence the total heat lost by
i he portion of the bar from x = « lt to x = I in a second is
* James David Forbes (18091868) was Professor of Natural Philosophy in
the University of Edinburgh from 1«J3 to i860.
VIII.]
CONDUCTIVITY BY PERIODIC FLOW METHOD
187
<j
Apc \w
dx
This will also be equal to the heat crossing the surface at x = x lf viz.,
since the bar is long enough so that its other end is at the room tem
perature and hence there is no loss of heat from that end. Equating
: i se two quantities we have
Kldd
P c
:d$
(f) = £*■
(13)
de
For calculating — the dynamical experiment is performed on an
actly similar specimen with the same exposed surface. The
observations are plotted in Fig. 7 which is selfexplanatory. The 8, x
and 0, i curves are drawn from actual observations while the values
of— corresponding to various values of are computed from the
0, t curve and plotted as indicated. Equation (1$) yields
— tan <p=P,
,■>.
where tan <£ and F are indicated in the figure. The area F of the
shaded portion can be measured by means of a planimeter and hence
K calculated.
Tliere are several, sources of error in Forbes' method and hence
the method fails to give accurate results. The specific heat docs not
aain constant at different temperatures as assumed by Forbes, Fur
ther the distribution of temperature inside the bar in the statical and
dynamical experiments are different. Forbes" method has been im
proved by Callendar, Nicholson, Griffiths and others.
rv. CONDUCTIVITY BY PERIODIC FLOW METHOD
10. Angstrom's Method. — The conductivity' of a metallic bar can
also be found by periodically heating and cooling a portion of the bar
and observing the temperature at different times at two points along
the bar. This method was first employed by Angstrom*.
In his early experiments a small portion of the bar was enclosed
in a chamber through which steam and water at any temperature could
be alternately passed. In later experiments the end instead of the
middle was heated. The bar was heated for 12 minutes and cooled
also for the same time, the periodic time being 24 minutes, Texnpera
* Anders Jonas Angstrom (18141874) was Professor of Physics at Upsafa.
He made important researches in heat, magnetism and optics.
188
CONDUCTION OF HEAT
[CHAP,
mres were observed every minute at two points along the bar by
means o£ two thermocouples. The mathematical analysis of this case
is somewhat complicated and will not be given here. We have to
solve equation (6) such that the solution is a periodic function of
time.
11. Conductivity of the Earth's Crust— The periodic flow method
is very suitable lor finding the conductivity of the earth's crust. The
earth's surface is heated by day and cooled by night. This alternate
heating and cooling effect travels into the interior of the earth in the
form of a heat wave {diurnal wave) and gives rise to the diurnal
Ltiona in temperature at points inside the earth's crust. Again,
the earth receives a larger amount o£ heat in summer than in winter
and this causes a second heat wave having the period one year (annual
wave) which is also propagated into the interior of the earth. Assum
ing the waves to be simple harmonic as a first approximation (the
annual wave.., in particular, departs considerably from this ideal state) >
: find how they travel into the earth. The problem is thai
bar pa iodi ally heated and cooled at one end and provided with a
i J  — ty *
A simple harmonic solution of (6) is
fl = ^** sin (orf+fi*+y) s . <  (14)
which gives the temperature fluctuations at any point x. This is
the equation of a
damped ogressive
wave and is graphically
represented in Fig. 8.
The wave moves for
ward with the velocity
u /5 whil its a ipii
tude on diminish
ing exponentially (given
by o ff~**) which is
shown by the dotted
curve. These fluctua
tions will be superposed on the mean temperature at any point which
will also diminish as we go farther from the hot end.
Tt. can be easily seen that the wavelength of the temperature
■wave is A = — Sor/jQ. Again, by substitution from (14) in (6) we get
—a Hfst • • • < 15)
Hence a=Jv}kT.
Now if a number of thermometers are embedded in the earth
at different depths, the progress of the temperature wave inside
the earth ran be investigated, and knowing the wavelength, the
diffusivity h can be calculated from (15), In the same way we cap
find the conductivity of any bar if we heat one end in a simple
harmonic manner as was done by King,
cAX
8L— Temperature wave at a particular instant.
VIII.]
CONDUCTIVITY OF THE EARTH'S CRUST
189
12. Applications,— The above mathematical treatment can be
utilised for solving the deal problem, viz., the penetration of
the dailv and annual changes of temperature within the earth's
crust. It will be easily seen fro I '.: that the range of ajnphtucK
R at any point is
J2=20«0V«/A?
and velocity of propagation
— ?= 2 j/f • ■ ■ < 17 >
and wavelength A= 2yV/*T. : •  ( 18 )
The "lag/' i.e, } the time which any temperature applied at the
surface talces to travel to a point x is
v 2 (/ rt'
,l.i.
We shall now apply these results to the propagation of heat in
earth's crust. Here for the daily wave
T = 24 hours — 86400 sec.
Taking h = 0.0049, the value for ordinary moist soil, we have
A — 73 cm.; v = 84 X '0 * cm. /sec. ; <* = 1/11.5.
Suppose the maximum temperature is 45°G at 2 p.m. and the
minimum is 25°C. just after sunrise, the amplitude of temperature
variation is, therefore, 20*C This variation will diminish as
e ~*iiL.i at a deptli x. For x — 10 cm., *■/»♦« = .42, for x = 30,
tf *ni.s_ j; an d for x — 100, **/"••= .00018. Thus at a depth o!.'
1 metre the temperature variations will be scarcely noticeable.
The slow velocity. of penetration of the daily heat wave must hi
been familiar to all observant minds in a tropical country. Here the
roof of the house is exposed to the scorching heat of the sun. and at
2 p.m. the temperature may be as great as G0°C. But this tempera
ture travels inward at the extremely slow rate of 9,1 X 10~ 4 cm./sec
or 6.4 cm. per hour in a mass of concrete (h — 0,0058) , Hence to
penetrate a wall depth of 30 to 40 cm. a period of 5 or 6 hours is
necessary. The inside of the room, provided the windows are shut,
therefore, reaches its maximum temperature at about 7 or 8 p.m.
when the walls become intolerably hot and begin to radiate. Most
:. must have experienced that it is found impossible to sleep
•rs at this time. The minimum at the top of die room is
reached at about sunrise. So the rooms are found to be cool from
..v£. to about 2 p.m. when the outside is hot.
Annual Wave. — Besides the diurnal fluctuations the surface of
arth is also subject to an annual period of 3G5£ days owing to
the different amount of surface heating in winter and summer. This
amount is variable in different countries, but in desert countries it
may amount to as much as 60°C in. the sun in summer and 0°C.
in winter.
190
CONDUCTION OF HEAT
[CHAP.
v.: ;'.
In the case of t he ann ual wave.
X = 73V 365 /' 1 — 73 X 191 CD1  = H metres
: 8.4 X JG~V^^ 5 c W sec  ~ 3 9 cm  P er da y
1_
220 '
Thus the annual wave will penetrate a depuh o£ about 1.2 metres
month, At a depth of x=\/2=7w. the times of the year
fl.6 1/3(35 ~2
» interchanged. The amplitude of temperature variation will he
reduced by the fraction e~™>™ — ,64 at 1 metre depth and bv
fiimt*» = ,011 at 10 metres depth. Thus the annual wave is able
to penetrate to a depth which is V365/1 = about 19 times greater
than the daily wave. Most of these conclusions have been verified
wall is made up
experimentally.
13. Conduction through Composite Walls.— If a
of a number of slabs of thicknesses x t , x 2 « . . and conductivities &i>
,v joined together, the amount of heat flowing per second through
an area A of the wall in the steady state is
a
*i
E t A
. ,;i.re the temperatures of the intermediate surfaces and
lr s u are the bemperaturea of Line end faces. This is because in
ite the same amount of heat, must flow through each
"sice
#i
* t 0, = (fcjJ
0a~ fl » — Q f~5
A « I
••■
and by addition
b** .  a ( jg ^
Therefore the heat, (lowing per second is
a:
*l_+ 3
(19)
+
K\A ' K 2 A
14. Relation between the Thermal a ad Electrical Conductivities of
WienemannFranz Law.— A table of thermal conductivities
is given on p. 200. The table shows unmistakably that all good con
ductors of electricity are also good conductors of heat, and even
before any theory was proposed, Wiedemann and Franz gave the
empirical law that the ratio of thermal and electrical conductivities
at a particular temperature is the same for all metals. Lorenz
extended die law and showed that this ratio is proportional to the
absolute temperature, viz., KfaT = constant, where K is the thermal
conductivity and a the electrical conductivity,
vm.
HEAT CONDUCTION IN THREE DIMENSIONS
39r
Drude explained this remarkable result by assuming that the
conduction of heat and electricity in metals takes place by means of
tree electrons. He even found theoretically the value of K/&T.
These conclusions have been experimentally verified by Jaeger
and Biesselhorst and by Lees. They obtained die value of K/vT
experimentally for various metals which are given in Table L It
will be seen that the value of K/qT remains practically contant for
all metals.
Table L— Values of J~X LO 8 
a i
From Lees'
experiment
Jaeger and
l ii: i.sclhorst
Metal
 170°C.
1.50
 ioo°c.
o°a
17 D C
10°C.
2.19
100°C.
Aluminium . 
1,81
2.09
2.1.3
2.27
Goppi
1.85
2.17
2.30
2.32
2.29
2.32
Silver
2.01
2.29
:'. "
2.33
2.36
Zinc
2.20
2.39
2.45
2.43
1
2.3,3
Cadmium
2.39
2.43
2.40
2.39
2.43
2.44
Tin
2.48
2.51
2.49
2.47
2.53
2.49
"J cm:
2.55
2.54
2.53
2.51
2.46
u\
Iron
3.10
2.98
2.97
2.99
2. 70
2.85
Brass
2.78
2.54
2/1.3
2.45
. ,
. ,
Manganin
5.94
4.16
3,41
3.34
3.14
2.97
A closer examination of the tabic shows that the value falls
off at low temperatures. Now at these temperatures both the
thermal and electrical conductivities are found to increase. Hence'
it follows that the thermal and the electrical conductivities do pot
increase in the same ratio when the temperature falls, the electrical
conductivity increasing much more rapidly. In fact, the latter
appears to become extremely large at the absolute zero, if the
metal is free from impurities.
15. Heat Conduction in Three Dimensions. — Till now we have con
sidered the flow of heat in one direction only, generally along bars
great length but small width and thickness. We shall now con
sider the conduction of heat in three dimensions inside an isotropic
body. Tt can he shown by an amplification of the ideas given in
sec. fi that for flow in three dimensions, the equation of conduction
is given by
ri&d , d*8 , d*9\ m
or
«*£
(20)
192
CONDUCTION OF HEAT
CHAP.
where h is the diffusivity. This is the Fourier equation ^ of hear
conduction in general and we can solve it when the initial con
ditions are given. Equation (20) is of great importance in &tu dy
ing problems on heat conduction*
V, Conductivity of Poorly Conductinc Solids
16, In finding the conductivity of poor conductors the sub
stance cannot be employed in the form of long bars or rods as was
the case with metals, for the heat loss from the sides would be
considerable compared with the heat actually conducted away
through the substance itself. For this reason the substance is
generally used in the form of thin plate, sphere or cylinder. To
this class belong all nonmetallic bodies. Cork, asbestos, clay,
wood, bricks, etc. are among the many substances of common
occurrence. The conductivity of these varies from 0,01 to 0.00008.
We shall now give some of the important methods of finding the
conductivity of these substances.
In the methods commonly employed for finding conductivity of
Soorly conducting solids, energy is supplied electrically to a separate
eating body, and the flow of heat through the experimental subs
tance investigated.
17. Spherical Shell Method. — The simplest case of heat conduc
tion in three dimensions is that of a sphere. If the source of heat, is
iced at the centre of the sphere the isothermal surf; ill be
1 surfaces described about, the centre. The method has been
employed ind others. Nfusselt's apparatus is shown in
80 cm in diameter and
in;;. pper. Inside it tcentrk with it, is another hoi!
sphere 15 cm, in diameter. The spheres can be split
into two ted. The space between die spheres is filled with
the material B under test such as
asbestos, powdered cork, charcoal
etc. An electrically heated body
D is placed at the centre of the
sphere inside C and electrical
energy is supplied at a constant
rate. Temperatures are deter
mined by means of thermocouples
:i Eh B at different distances
from the centre and alonir one or
more radii. Knowing the elec
trical energy spent and the radii
of the shells, the conductivity can
be calculated.
We shall solve the problem from elementary principles. On
'lit. of the symmetry about the centre the isothermal sua i
Clerical. The flux of heat across a spherical surface of radius
r outwards in unit time is — KAfffi t. This must equal the amount
d r
Fig, 9, — KTosselt's apparatus.
VIII. J
CYLINDRICAL SHELL METHOD
195
of electrical energy (J supplied per second to the heating body.
Hence*
Or
4 77 A
dr
dr
Since () is independent of r we have on integrating (21)
8 =
Q,
4tt/l"
4+*
(21)
(22)
where A is a constant of integration.
Tf the two surfaces of the shell of radii r lt r 2 acquire temperatures
8 l3 &£ in the steady state, we have
4irR' r t ^*' * 2 4wK
 4 A,
nee
0.=
4fliT(^fl Jf a
Knowi g 0i,02 and r x , r 2 as in the above experiment the con
ductivity K can be calculated. Again, solving the simultaneous
equations for C> and A, and substituting these values of Q. and A
in (22) , the Cempeiature distribution in the material can also be
found, and is given by the expression
« = r 1 r [Mrt«+ ft ~' JV 4 .* ■ (241
18, Cylindrical Shell Method. — Let us consider the radial flow
of heat in a cylinder which has an electrically heated wire along its
axis. By suitably transforming equation (20) we can get the equation
for this case. We shall, however, as before, deduce it from elemen
:>nsideraliom. Since the cylinder is symmetrical about its axis,
the isothermal surfaces are cylindrical, The amount of heat {)
Rowing per second across an isothermal surface is
ft . 2wrfjr=. p
<.••
where I is the length of the cylinder. Now (J must remain constant,
being equal to the electrical energy supplied per second. On integrat
between the inner and outer' radii of the cylindrical shell with
the corresponding temperatures B it & 2 , we have
or
2W" 8 x 0 t
(26)
The temperature & at a distance r can be shown to be given by
9 = log (rJr ) [ ($1 l0g r *~ $ * l0g ^ _ ^i^ lo « '") ] ■ •
*We can also proceed as in sec. 6.
13
194
CONDUCTION OF HEAT
CHA '■
distance from
en nation (2,6)
II lias devised a method of determining conductivities based on
this methi n er o£ the material is formed by filling
space between two hollow concentric cylinders with that material.
Heat is supplied by a wire carrying an electric current along the axis
oE the cylinder, and when the' steady state is reached, temperatures
at two points within the material are observed, as well as their
the axis. Knowing  the electrical energy spent,
gives the conductivity.
simple laboratory expc
rimeiit based on this method
may be devised for finding fch
conductivity of rubber and
glass in the form of a tube. For
rubber the arrangement shown
iu Fig. 10 is most convenient.
Steam from the boiler A tra
verses through a rubber I
H t a length J! of which is
mersed in a weighed amount of
water contained in the calori
meter C. The radial flow of
heat Q_ from the tube to the
water in C is given by
Fig. 10, Apparatus for finding
conductivity of rubl
log.
(28)
re $i n the temperature of steam, and {.), the mean of
the initial and final tern] ■ of the calorimeter. This heat can
in temperature of the calorimeter and its
id hence equating these two quantities we get
it*.
For ;emenl shown in Fig. 11 is convenient.
is tube C has a spiral wire along its axis and is surrounded by i
mi jacket, while a steady current of water flows through the tube.\
The method of calculation is similar to the preceding one.
Walt,
Sieam
W ate *
Fig. lb— Apparatus for finding conductivity a£ glass.
19. Lees* Disc Method* — Another convenient method for finding
conductivity of a very bad conductor has been given by Lees
♦For experimental del Worsnop and Flint, /Vc ■'■. asks.
DHL]
LEES' DISC METHOD
J 95
Steam
in which the substance is used in the form of a thin disc. The
apparatus (Tig. 12), due to Lees and Charlton, consists of a cylindrical
steam chest A, the bottom of
which is a thick brass block B in
which a hole is bored for inserting
a thermometer T lf The substance
S (shown shaded), En the form of
» a circular disc, is sandwiched
between the block B and a second
cylindrical brass block C, the latter
carrying a second thermometer
I s . The I^ck C is suspended
m a retort stand by threads so
that its top face is horizontal.
The radius and the thickness d
of the specimen are first measured.
Fig. 12— Lees' disc method
f f bad conductor.
The apparatus is then set up as shown in Fig. 12 and steam is
d through A for a pretty lone time until the readings in the
thermometers T 3 and T 2 are steady. These temperatures can be
received by it from conduction through the specimen must be equal
to the rate at which heat is lost by it by radiation from the sides
and the bottom surface of C. The former is given bv
GJM&^&, . . . (29)
where A' is the required conductivity and A the area of the disc. The
latter is determined in the following iva; ;
The block C h alone heated separately by a bunscn burner until
its temperature rises to about !0°C above 2 . It is then alone sus
pended with the specimen placed on the top and allowed to rool.
Its temperature is noted at regular intervals of time until it cools to
about 10°C below g 2 and a cooling curve drawn, From the tangent
to the curve at 8 2 , the rate, of cooling  f at 0., is determined. The
rare of loss of heat by cooling is
a— 'GL
. (30)
where m and s denote respectively die mass and the specific heat of
the block a Combining (29) and (30) , K can be found.
In die above we have assumed that no heat is lost from the
curved surface of the specimen, and consequent! v the specimen must
ery thin.
The disc method can also be used with electric heating when a
modified arrangement is necessary. A thin plate of the experimei]
J 96
CONDUCTION OF HEAT
substance is placed between two copper plates, A coil o£ insulated
wire is place above die tipper copper plate and held down by another
copper plate as explained later in § 22. Electrical energy is supplied
to the coil and steady state obtained. The amount of heat passing
through the experimental disc can be obtained from the electrical
energy supplied and the loss of energy due to surface emission.
Knowing cue temperature of the copper plates the conductivity
be calculated. Alternatively, the twoplate system can be employed
by placing above the heater coil another sandwiched experimental
plate, exactly similar to the system below the heater coif. In this
arrangement it is not necessary to know the radiation loss as all the
electrical energy is transmitted through the two specimens.
CONDUCTIVITY OF LIQUIDS
20. The determination of the conductivity of a liquid is
complicated by the presence of convection currents. If we heat
a column of liquid at the bottom, the liquid at the top receives
heat both hy conduction and convection. The laws governing convec
tion currents are complicated] hence it is preferable to eliminate them.
This is accomplished either by taking a column of liquid and heating
it at the top or by taking a thin film of liquid.
21. Column Method. — The column method was employed by
Dcspretz long ago. The liquid at the top was kept heated and tem
peratures axis ot the column were observed by mercury
He Found that Fourier's equation already derived for
a bar hold? true in this case riiso, and hence the conductivities of two
Liquids caj with the help of equation (11). Weber sur
fed the column with a guardring and used as the source of heat
an electrically heated oil bath. The bottom of the column was cooled
copper plate standing in ice. The heat conducted away was
found from the amount of ice melted. Knowing the temperature
bution along the column the absolute conductivity can be calcu
lated in the same manner as for mercury in Berget's experiment.
22. Film Method,— This was employed by Lees>, Mihier and
Chat Lock and by Jakob, We
shall describe the apparatus
iiied by Lees which is simply
a modification of his disc
method for finding the con
ductivity of poor conductors.
The liquid L under test is
enclosed in an ebonite ring E
(Fig. 13) and placed between
copper blocks, C^ C.,. To de
termine the quantity of heat
flowing through the liquid a
glass disc G is inserted, above
which another copper plate
Above C 3 and insulated from
Fig, 13. — Lees 1 apparatus for finding
the rouductivity of liquids.
C 3 is cemented with a layer of shellac.
CONDUCTIVITY OF GASES
197
it by mica is placed a flat spiral coil of heating wire W which is held
down by another copper plate C. The whole pile is varnished and
enclosed in an airbath. Temperatures are recorded by thermocouples
soldered to the faces of the copper plate. The calculations can be
easily made.
Let Si, S s , Sg, S a denote the emitting surfaces of CY, C 2! C s
and G respectively, h their emissivity and T,., T„, T & the temperatures
i I' the copper discs. The heat passing through the middle section
of G is by definition equal toai£T G (T a  F a )/d . Of this the amount
9 ' —^o — 2 i s I° 5t fry tne lower half of G by radiation from the sides
and similarly S 3 h T,. is lost by C s . Hence the heat transmitted
through the liquid and the ebonite is equal to
AK t
T 9 r t
— SqH ' '— , —  — $tkT s
and thus must evidently be given by
: B(T,T i ),
some constant giving the transmission of energy through
the ebonite E for unit difference of temperature. Equating these two
expressions we get K& in terms of K G provided B is known. B can
be determine! ::rking with air whose conductivity is known.
23. Hotwire Method. — Goldschmidt employed the "hotwire"
method of Andrews and Schleiermaeher, the theory of which is dis
cussed in sec. IS. The liquid was contained in a silver capillary Lube
2 mm. in diameter heated by a wire running along its axis,
CONDUCTIVITY OF GASES
24, The determination of the conductivity of gases is difficult for
the phenomenon is always accompanied by radiation, and sometimes
by convection currents also. Kundt and Warburg showed that the
rate of cooling of a thermometer immersed in air remained constant
for pressures lying between 150 mm. and 1 mm. Hence the effect
of convection currents is negligible in this region and heat is lost only
by conduction and radiation. To determine the radiation loss, the
air was exhausted as completely as possible, and then the rate of cool
ing was found to be independent of the size of the enclosure, showing
diaf the effect of conduction was negligible and heat was lost only
by radiation. Subtracting this radiation" loss, we get the heat lost hy
conduction alone. For finding the conductivity of a gas at pressures
higher than 150 mm. the gas may be exhausted to this pressure (be
tween 150 mm. and. 1 mm.) and its conductivity determined. Now
since the conductivity of a gas is independent of the pressure (see
Chapter III, Sec. 24) this will give the required conductivity. Another
198
COXDUCriON OF HKAT
[CHAP.
Fiff. 14. — A '_:■•.: 'd'atus to sllf>w the
better conductivity of hydrogen
procedure consists in taking a thin, film of gas and healing it at the
top when convection will be absent
A very simple experiment
described by Andrews and Grove
allows qualitatively that, hydro
gen is a far better conductor of
iat than any other gas, A fine
latinum wire was supported in
side a glass tube (Fig. 14) which
could be filled with any gas.
. wire could be heated by an
electric current and made to
glow. Two such tubes were
; ranged side by side, one filled
with air and the other with
hydrogen. The same electric:
current was allowed to I
through both the wires. The
wire in the airtube can be
glowing while the wire in the hydrogen tube does not glow at all. The
tss tube containing hydrogen also becomes hot. Heat is very quick
ly conducted awa '■■ and hence the wire is not raised to
the temperature of incandescence. Replacing hydrogen with air res
totes the incandescence.
[<e important methods of finding the a i xmductivity of a
■
Method, (2) Film Method.
25, Hotwire Method. — This method, first given by Andn
emi. acher lor determining the absolute conducti
vity electrically heated vertical wire is surrounded by
a coaxial c filled with the experimental gas. The temp.
e of the wn:e is known from its resistance, and the amount of heat
flowing across the curved surface of the cylinder is found from the
rate of energy supplied to the wire. The conductivity can be calculat
ed from equation (2ft) where r t , r s now denote the radii of the wire
and the tube respectively. In this arrangement convection is very
much minimised sine:: there is no temperature gradient in the vertical
irei ion, Radiation losses and whatever convection losses remain,
eliminated as pointed out. in sec. 24. Correction must, however,
be applied for loss of heat by thermal conduction along the leads
supplying the electrical current. The method has recently yielded
■• accurate and reliable values for a large number of gases.
26, Film Method, — The film method, originally .used by Todd,
has been employed by Hereus and Laby. The principle underlying it
is th^ same as in Lees' experiments (sec. 22) . The thin film of gas
under test was enclosed between two copper plates B and C (Fig, 15),
viu.'l
CONDUCnvrTY OF GASES
199
the latter of which was cooled by a current of water. The upper plate
B was made up of two sheets of copper clamped together with a heat
nl between them. To prevent loss of heat by radiation from
the upper surface of B, there was another plate A above it at the same
temperature and a guardring D surrounding it. On account of the
latter the flow of heat from 1> to C was linear. The plates A, B, D
separate heating coils and thermocouples and were kept at
lame temperature. The thermocouples were formed by attaching
untan wires to the copper plates, each plate having a copper
lead also attached to it. The plates were accurately ground and silver
plated. The whole apparatus was made airtight by a ring of rubber
clamped to A and C by steel bands. The temperature of A was gener
Ebanitc
BcaiS
iV iier
. IS.— Apparatus of Hereas and Laby.
ally kept a little above that of B in order to eliminate :; ability
onvection. This, however, necessitated a small correction. The
radiation correction was determined by a separate experiment on a
silvered Dewar flask and was only about 5 per cent. Convection
effects arc: entirely absent since tlie gas is heated at the top, The
energy spent in B was known electrically and subtracting from if the
lost by radiation, the heat transmitted to C by conduction through
the air film wis found. Knowing the temperature of C the conduc
tivity of the gas can be calculated from equation (1) .
27. Results, — The thermal conductivity of a number of sub
stances is given in Table 2, p. 200. The value of K is given in
calorie cm. 1 sec.? C X *
It will be seen from the table that silver is the best conductor of
heat (K — 1) and copper comes next. The conductivity is less for
liquids and least for gases. The conductivity of gases is extremely
(of the order of 1CH 3 ) .
We have: already considered on p. 85, Chap. Ill, the relation
between thermal conductivity and viscosity of gases, as well as the
variation of the thermal conductivity of gases with pressure.
200 lUCTlON OF HI ■ ■■. : [CHAP,
Table* 2. — Thermal conducti of different tances.
Substance
Metals (0°C.)
Aluminium , , j
.
Cadmium .  !
0.23
Copper
0.93
Iron (pure)
0.16
Lead
0.085
Mercury
0,02$
Nickel
0.11
Silver
1.0
Platinum
0,17
Tin
0.155
Zinc
i. :.r:.<
Asbestos paper
Cardboard
Cork (p=.16)
Paper
Ebonite
Mica
Paraffin wax
Pine wood
Rubber
Alloys (0°C.)
Brass
an
Mauganin
0.05
0.06
Poo\
1 lint glass
2.5x10
2
Liquids
Water (24°)
Alcohol (25 D )
Glycerine (25*)
0.6k 10"
0.5
0.11
0.3
0.42
1.8
0.6
0.4
0.45
14.3XJ0 4
4.3
jes (0°C.)
Helium
Argon
Air
<:Ll
Carbon dioxide
10 ■*
3.89
5.40
5.63
8.07
28. Freezing of a pond.— An interesting example of conduction
across a slab of varying thickness is provided by the phenomenon of
freezing of water in ponds and lakes during winter. When ice I
to form on a pond, the bulk of the water in the pond is at aboui
while the top layers are at 0°G, and the cold air above abstracts the
latent heat from a narrow surface layer at 0°C. The subsequent
growth of rile ice layer requires that the necessary abstraction of
latent heat take* place by conduction through the layer of ice already
there. Assume that, the thickness of the ice layer already formed is
z, and die temperature of the water below this layer is 0°C. Then
for a further freezing of the layer of area A and thickness dz in the
time dt 3 the quantity of beat
Q = Apdz.L
Taken partly from Landolt and Bernstein, PhysikalUhChemischen Tabelten
and partly from Kaye and Laby, Tables of Physical and Chemical Crttwttents,
CONVECTION Ol 7 HEAT
201
vii i.]
uu.l travel upwards by conduction through ice of thickness t, wner
P = density of ic& and L — the latent heat of fusion Hence
where K = thermal conductivity of ice, and £ is the temp era Cure of
the aiT above the pond. Equating we get
pL
which on integration yields
P L
Lhe constant of integration being zero as z = at * = 0. Thus the time
10 obtain a given thickness is proportional to the square ot ttie
thickness.
Exercise.— The thickness of ice on a lake is 5 cm. and the tem
perature of the air is 2G°C. Find how long it will take for the
thickness of the ice to be doubled. (For ice thermal conductivity is
0005 cal cm 1 sec. 1 C 'C~\ density = 0,92 gm./cx,, latent heat = 80
cal./i i
[\Ve have
*KSH
IpSa'* pL
t= = 2 KB
10'— 5* 092x80
~2 0405x20
= 27,600 sec = 7 A 40 rtt .]
CONVECTION OF HEAT
29. Natural aad Forced Convection*— We have already stated in
§ 20 and § 24 that, there will be transfer of heat by convection in fluids
unless proper precautions are taken. Convection is the transference
of heat by heated matter which moves carrying its heat with it. Thus
it can take place only in fluids. Free or natural convection always
takes place vertically and is caused by gravity as a consequence of
the change in density resulting from the rise in temperature and con
sequent expansion. ' In forced convection a steady stream of fluid
is forced past the hot body by external means. Stillair cooling is
ural convection ; ventilated cooling in a draught is forced convec
tion.
The theoretical treatment of convection is rather complicated
though the problem of forced convection is a little simpler. Never
theless convection is of great practical importance. The land and
202
CONDUCTION OF HEAT
[CHAP.
sea breezes, the trade winds, the tall of temperature with height in
the atmosphere (discussed fully in Chap, XIT) are all examples of
convection on a huge scale in nature. The ventilation of rooms and
the central heating of buildings in winter arc some examples of forced
convection in e veryday life.
30, Natural Convection, — 'When a heated body is cooled in air.
all die three methods of heat transference are acting sirimltaneoush.
But the air is a very poor conductor of heat, and radiation is import
ant only for large differences of temperature ; thus the chief means of
heat loss is convection. The mechanism of this heat loss is easy to
understand but the derivation of a theoretical formula is extremely
difficult. The problem is therefore best studied experimentally or
by a recourse to the method of dimensions.
In [lie film theory of cooling by convection the mechanism of
heat loss is somewhat as follows :— It is believed that the whole sur
face of the cooling body is covered by a thin layer of stagnant fluid
adhering to the surface. In natural' convection this film is perma
nently present while in forced convection this is being continuously
wiped off and renewed, Thus in natural convection the heat has to
flow across this film of air and the amount lost will depend upon the
thermal conductivity of the air and the temperature difference between
the body and the air. The heat transmitted will raise the tempera
ture of the air > convection which will he opposed by via •
forces. Thus the heal loss will depend upon the specific heat, expan
nt and viscosity of the air.
experiments on natural convection was
edit. They observed the rate of cooling oi
ii :i large coj iper globe surrounded by a hath at a fixed
riments were first, done with the globe eva
when the heat loss w is due to radiation alone and the rate ■■
fall of temperatui Bound to be
5*c^>.
(31)
where ' a constant depending upon the nature of the surface and
Q, 8 the temperature of the thermometer and the enclosure res
pectively. Next the globe was filled with different gases at different
pressures, and rate of cooling observed. Subtracting from this the
radiation loss, the loss due to convection alone was computed and
found to he
^^nipHd6^*.
m
where p is the pressure of the gas and nij c are constants depending
upon the gas, Thus the rate of loss of heat due to natural convection
can he written in the form
 f= k(do )^
VL1J.
CONVECTION OF HEAT
203
where k is a constant depending upon die gas and its pressure, and
5/4 ha^been written instead of 1.233. This Js called the fivefourths
power law for natural convection which can be also deduced theore
tically.
For forced convection experiments show that the rate of loss
of heat is proportional to die temperature excess (<?0 o ) . It will be
thus seen that Newton's law of cooling holds for forced convection
even for large temperature differences but is not true for natural
convection. As shown in Chap. XI it holds for radiation provided
the temperature difference is small.
Books Recommended
1. Glazebrook, A Dictionary of Applied Physics, Vol. T.
2. Ingersoll and Zobel, Mathematical Theory of Heat Conduc
tion (Ginn, 1913).
3. Carslaw and Jaeger, Conduction of Heat in Solids (10 :
Clarendon Press, Oxford.
CHAPTER IX
HEAT ENGINES
L Introduction to Thermodynamics*— Thermodvn amirs is liter
al I j the science that discusses die relation o£ heat to mechanical
energy. Rut m a broad sense, in comprises die relation of heat to
other forms of energy also, such as electrical and chemical energy,
it energy, etc. The principles of Thermodynamics arc very gem
m their scope, and have been applied widely' to problems in Physics,
Chemistry and other sciences. The theory of heat engine* r ; Q
an integral part of the subject, and as the 'early developments were
largely in connection with the problem of conversion of heat enemy
S S eC ^ 1C W ?™' we sha . 1] ^gin the study by devoting a chapter
to the Theory of Heat Engines,
HEAT ENGINES*
2. The progress of civilisation has been intimately bound up
with mans capacity for the development and control of power
History tells us that whenever man has been able to make a great
discovery leading to a substantial increase in his power, a fresh epoch
m civilisation began, *
he present ag sometimes been styled as the 'Steam Aee'
1 influence exerted by the invention of the
^ of human proves, Tn this chapter.
u make a brief survey of this "great event."
present time we know that Heat is a kind of motion
r motion disappears it reappear, as heat, and experiments
hat 1 calorie of heat is equivalent to 4.18 x 10* ergs of work.
e question naturally arises : "Can we not reverse the process? Can
we not by some contrivance, convert heat which is in so much excess
;»t m, to useful work?" This is in fact the function of heatengines
to wo r r arE g ° ing t0 They "* contrivailces to convert heat
"N»m£ J? wJHSft to «*» to » u jw« Profoundly ignorant of the
irfrhlffnl 4t he T thC Pr0 ? em did not F* 6 * *** to them
! r t,^ Z hey \ bwever ' ? bseiTed that B«wDy when bodies
verTbodv C ?' P P ° Wer " D ° ay take three exa Wfe familiar
t™ ,!; When WaL T i 5 ^ iled in a dosed ketdc > rhe lid is blown off
place to acknowledge our grateful thanks to the author.
TX.j
EARLY HISTORY OF THE STEAM ENGINE
205
Z When gunpowder or any explosive is exploded a sudden im
pulse is created which may be utilised for throwing stones, cannon
balls and for breaking rock.
3. High velocity wind can be made to do work, e.g., from early
times sails have been used for the propulsion of ships, lor driving
mills (windmills) . We know that such winds arc due to intensive
heating of parts of the earth's surface by the sun.
The three illustrations chosen above have served as the starting
point for three different classes of engines which convert heat to work,
viz : (I) the steam engine widely used for locomotion and in in
dustry, (2) the internal combustion engines used in motor cars, aero
planes, and for numerous other purposes, (8) the windmills and steam
and gas turbines.*
Many of the principles utilised in these engines art mute common
to all classes, and we shall begin by describing the evolution of the
steam, engine. Though the mechanical details" are outside the scope
of this book, an elementary discussion is included for the sake of
completeness and continuity of treatment.
3 Early History of the Steam Engine. — The earliest record of
human attempt to make a heat engine is found in the writings
Hero of Alexandria, a member of the famous Alexandrian school of
philosophers (BOO B.C.— 100 A.D.) which included such famous men
of science of antiquity as Ptolemy (astronomer) , Euclid (geometer)
and Eratosthenes (geographer) . Hero describes a scientific toy in
which air was heated in a closed box and allowed to expand through
a pipe into a vessel below containing water. The water was thus
forced up through another pipe into a vertical column producing an
artificial fountain. There was, however, no suggestion to employ it
on a large scale. In 1606, about two millenia after Hero, Marquess
Delia Porta, founder of the Neapolitan Academy and one of the
pioneers of scientific research in Europe, employed steam in place
of air in Hero's experiment in order to produce a fountain. He also
suggested that in order to fill up the vessel with water, it may be
connected by a pipe to a water reservoir below. If the vessel filled
uo with steam be now cooled with water from the outside, steam inside
will condense, a vacuum will be produced, and water will be forced
up from the reservoir, replenishing the vessel again. This principle
was utilised by Thomas Savery in 1698 to construct a waterpump in g
machine which is described below. He was the first man to produce
a commercially successful steam engine which was extensively used
for pumping water out of mines, and supplying water from wells.
The principle utilised in Lhis engine is illustrated in Fig, 1 (p.
206) . V is a steam boiler. A, B and" C are valves, The operation
takes place in two stages * —
♦Recently during the second world war a new type of engine based on the
rocket principle was developed in Germany and Italy. In these a high velocity
jet of air escapes at the rear of the machine which on account of the reaction
thus produced moves forward with tremendous velocity,
T
206
HEAT ENGINiS
j ( ■. i \r
ix.l
J AM liS W>
Fig. 1. Principle cf Savory's
(a) B is kept, closed and A. C are open, Steam passes from V
to P and forces tlie water Eg
(b) A and G are closed, and
B opened. Cold water is sprink
led on P, This condenses steam
in P, a vacuum is created and
water is sucked up from the pit
E to P, After this the operation
(a) may be again performed and
a fresh cycle begun,
Savery*s engine could not
suck water through, more than 34
Irt'i,, but it could force up the
water to any height. In fact,
be sometimes forced up water to
a height of 300 feet. This means.
that he used high pressure steam
up to 10 atmospheres. This was
a risky procedure though Papin had shown about 1680, how the risk
in using high pressure steam could be minimised by the introduction
of the safety valve.
Papin, a French settler in England, had discovered a method of
softening bones by boiling them in a closed vessel under pressure.
as we know, raises the boiling point of water to about 150°C
and makes the watei a very powerful solvent, Papin invented the
boiler to re vent his vessel from
being blown up by high pressure steam. This is shown in Fig. 2.
valve consisted of a rod LM
ted at I. and carrying a weight N at the
nd. Tt pressed down the valve P which
exactly fitted the top of the tube HH leading
from the inside of the boiler. Whenever the
mii pressure exceeded a certain limit, it
forced up the valve P and the excess steam
would rush out. By adjusting the weight of
or its distance from L, the maximum steam
isure could he regulated at will.
4. Newcomen'3 Atmospheric Engine* — The
next forward step was the invention of Newco
men's Atmospheric Engine which was designed to
pump out water from mines and wells, and was
in practical use for more than fifty years. This
ie is interesting from the historical point of view since it directly
led to the grear inventions of James Watt, and it employed for the
first time, the cylinder and the piston, which has been a feature ol
steam engines ever since. Fig. 3 illustrates the Newcomen Engine.
Fig. 2. iPap&i'g
Safety Valve.
A is the cylinder, T is the piston suspended by a chain from the
lever pivoted to masonry works. The other arm of the lever carries
the piston rod W of the water pump which goes into die well. There
is a counterweight M to balance the weight of the piston T. The
problem is to move the piston T up and down.
This was achieved as follows : — Starting with the piston T at
the bottom of A, steam is
introduced from the boiler
B which forces the piston up
till it reaches die top. The
steam is shut off by the tap
1), and cold water sprayed
through F which condenses
the steam in the cylinder.
Vacuum is produced inside
the cylinder and consequ
ently the atmospheric pres
ide forces down the piston,
I J is again opened and a
fresh cycle begins. The
water in the cylinder A
drains out through a side
pipe.
For closing and open ing
the valves automatically, a
allel motion guide was
provided which carried
Fig. 3. — Nciwcomeir'., ALii::.ii.u. lie
Engine.
mechanism lor automatically operating the valves. The story goes
that the invention was due to a lazy boy who was employed to close
and open the valve by hand, but who tied a parallel rod to the swing
arm of the lever, and connected it bv means of cords to the valves,
and leaving this rod to do his work enjoyed himself all the while in
playing. Whatever may be the origin, the parallel guide has been a
permanent feature of steam engines ever since.
In the Newcomen Engine, the useful work is done by the atmos
pheric pressure while steam is only employed to produce vacuum,
hence the name atmospheric engine. It is easily seen that it is very
wasteful of fuel.
5. James Watt. — James Watt is commonly credited with the
discovery of the steam engine. The circumstances which directed
his attention to steam engine are pretty well known. He was an
ingenious scientific instrument maker at Glasgow, and in 17G.5 he
asked by the professor of Physics at the Glasgow University to
: I .:• a mien Engine belonging to them which had never worked
well. Willie i : : i d in the' repair of this machine, the idea occurred
to him that the Newcomen Engine was awfully wasteful of fuel, and
being of an inventive temperament, he began to ponder and experi
ment on the production of a better type of machine, He was thus
f ENGINES
CHAP
led to a series of investigations and contrivances which gave the steam
engine its present form and rendered it a mighty factor in the onward
march of industry and civilisation. We are describing some o!
inventions below.
6. Use of a Separate Condenser,— Watt observed that a large
part of the expansive power of steam is lost on account of the fact
that the cylinder is alternately heated and cooled. The expansive
power of steam depends upon its temperature. Now when the steam
enters the cylinder, which has been previously cooled to create a
vacuum, sonic heat is taken up by the cylinder in becoming heated
and is not converted into useful work. The temperature ot steam
falls and its expansive power is diminished. Another disadvantage in
using the cylinder as condenser is that cold
water entering the cylinder becomes heated
and exerts appreciable vapour pressure.
thus preventing the formation of a goad
vacuum. The "problem was to cbndensi
the steam without, cooling the cylinder.
Watt achieved this by the use of a separate
condenser.
The principle of the separate conden
ser is illustrated in Fig. 4. A A is the
cylinder in which the piston P moves to
and fro. The piston is provided with a
; i n he PQ carrying a. valve Q at the
nser  end such that Q allow* steam to go out
,ni is closed by the atmospheric pressure when there is vacuum in
Eng with the piston P at the bottom of the cylinder, R r
j are and the part of the cylinder above P is filled with
nit the air and residual steam through Q. Then S
id T opened. The steam ■ Lwn into the condenser C
which had been slv evacuated by the pump B f and is there
ilensed by the cold water surrounding the condenser. Come
ly a vacuum is produced above P and steam from below pushes
die piston P upwards, doing work on the weight. W Then T is
closed. S opened and P is drawn down to the bottom by W and the
cycle begins afresh. The pump D serves to remove the air and water
produced from steam in G.
To keep the cylinder hot, Watt further surrounded the cylinders
by a steam box and wood. Nowadays the cylinders are jacketted
tth asbestos or some badly conducting substance, and then covered
with thin metal sheets.
7. The Doubleacting Engine.— In the Newcomen engine we
have seen that the atmosphere pushes down the piston. Shortly
afterwards Watt employed steam instead of the atmosphere to pull
the piston down. The raising of the piston in the subsequent stroke
was brought about by a counterweight attached to the other arm of
To face p. 208
Watt
(p, 207)
James Watt, bora in Scotland in 1736, died in 1819. His important
work is the masterly perfection of the steam engine which
Increased the powers of man ten times and ushered
the 'Industrial Revolution."
IX.]
t.ilLIo.VlIOX OF EXPANSIVE POWER OF STEAM
m
Cabnot (p. 213)
Nicolas Leonard Sadi Carnot, bom on June 1, 1796 in Paris,
died of cholera on August 24, 1832, He introduced
the conception of cycle, of operations for
heat engines and proved that
the efficiency of a reversible
engine n maximum
the beam. For these operations lo be possible the upper end ol I
cylinder must be closed. Watt achieved tiw ant o a steam
efat stuffing box which is full of oily tow. This is kept Ughtly press
ed 3 against trie piston so that the piston can move through the cover
without loss am. This was the socalled smgkactmg
Wat i Watt, however, soon realised that in this engine no
done'hv steam dining that stroke in which the piston was raised up
by the "action of the counterweight. He saw that the power could
be a mately doubled if during this useless stroke* steam is aa
mitted to the lower side and the upper side is connected to weefflj
i ;er. This is achieved in the doubleacting engine, invented by Watt,
with the aid of a number of valves, A modern doubleacting cylinder
. is shown in Fig, 5. The cylinder has ports or holes
\, B, near its each end and between these lies another port E leading
the exhaust or condenser. To the cylinder is fastened the
steam chest C containing the Itslide valve S. Steam from the boiler
enters the steam chest at the top. In the position (a) steam enters
the cylinder through the port B and pushes the piston to the left,
thereby driving the steam in front through A to the exhaust *,. AS
the piston moves to the left the slide valve moves to the right and
. 5.— Doubleacting cylinder with slide valve.
closes both the ports A and B For a time, and later when the i
v.* the extreme left position, B is closed and A opened. _ Steam
then enters through A forcing the piston backward and > ng the
The double
kinds of steam
then enters through A forcing the p
steam in front to the exhaust. This is shown at (b) ,
acting engine is now universally employed in all
engines. ,,.,, r
The timely action of the slide valve is adjusted by means of an
entric wheel attached to the moving shaft (see Fig. 7).
powerful engines as in locomotives the slide valve is often repla<
by a piston valve which is very similar.
8. Utilisation of the Expansive Power of Steam.— Watt's another
at inventi .the socalled expansive working of steam.
saw that if steam is allowed to enter the cylinder all the time
n is movi • tfards, the steam pressure in the cylinder wih
„e the sac i the boiler and though we get a powerful stroke,
the expansive power of steam is not utilised If, however, the steam
14
210
HEAT ENGINES
CHAP.
is cut off when the piston has moved some distance, the piston would
complete its journey bv the expansive power of steam, whose pressure
will in consequence be reduced to almost that of the condenser. Thus
more work is' obtained from the same amount of steam by allowing
the steam to expand adiabatically and hence the running of the
machine becomes considerably economical. It is thus of great
advantage to use high pressure steam.
It was mentioned in the last section that the slide valve closes
bo lit the ports A and B when the piston has moved some distance.
From this instant to the end of the stroke the steam is allowed to
expand adiabatically.
9. The Governor and the Throttle Valve.— Another simple but
very useful invention of Watt was that of the governor. This is a
piece of selfacting machinery which controls the supply of steam
from the boiler into . . cylinder, and ensures smooth running of the
engine at a constant spe
Watt's governor is shown in
Fig. 6. S is a vertical spindle
which is made to revolve by
means of gearing from the en
gine shafts. It* speed, there
fore, rises or falls with the
engine speed. It carries a pair
of heavy balls which are fasten
ed to S by rods pivoted at P.
The balls rise on account of
centrifugal force as the spindle
rotates, and as they do so they
pall down a collar C which slides
smoothly in the spindle S. The
ne end of a lever L, pivoted at Q. The other end
p in the steam ipe called the throttle valve. As
ulled down, thi ■ Live tends to close the tap, the steam
a] md the engine speed falls. If the engine speed is too
diminished, the balls fall down, C is pushed up, and the throttle
valv. tting more steam, and the speed goes up. Thus the
governor automatically regulates the speed at which the engine runs.
Improved forms of governors are now employed in engines.
10. The Crank and the Flywheel— Watt was the first to convert
the toandfro motion of the piston into circular motion by means o£
the connecting rod and the crank. Thus the steam engine can he
made to turn wheels in mills, work lathes and drive all kinds of
machinery in which a rotary motion is needed.
The connectingrod R and the crank C are shown in Fig. 7
at (a). The crank is a short arm between the connectingrod
ant! the shaft S. The connectingrod is attached to the piston rod
consequently takes up the to andfro motion of the latter. As
ingrod forward it pushes the crank and thereby
rotates the shaft S. In the return stroke the circular motion is.
(Ill: t
IX.]
MODERN STEAM ENGINES
211
I'iij. 7. — Crank, Eccentric and* Flywheel.
completed. There are., however, two points in each revolution when
the connectingrod and the crank are in the same line and the pis
ton exerts no turning moment.
These are called the 'dead
centres/ At two points when
the crank is at right angles to
the connecti ngrod the torque
is maximum. To prevent the
large variation in the magni
of the torque producing
variations in the speed of the
shaft during a single revolu
tion a big flywheel F, shown
at (c) s is attached to the shaft.
The flywheel on account, of its
large moment of inertia carries
the crank shaft across the
dead centres ', in fact, it ab
sorbs the excess of energ) sup
plied during a part of the half
revolution and yields had I
same in the remaining part of
the halfrevolution when less
energy is supplied. Thus the flywheel acts as a reservoir of energy
which checks variations during a single stroke, while the governor
prevents variations from stroke to stroke.
Another mechanism to convert the toandfro motion into cir
cular motion or vice versa is the eccentric, shown at (b) , Fig. 7.
It consists of a disc mounted off its centre on the shaft S and sur
rounded by a smoothly fitting collar to which the rod is attached.
The behaviour is as if there was a crank of length SC. . Such an
eccentric is mounted on the shaft carrying the flywheel (shown at c)
and works the slide valve. The effects can be properly timed by
suitably mounting the eccentric on the shaft.
The essential parts of a simple engine are shown in Fig. 8.
They will be easily followed from the figure.
11. Modern Steam Engines. — Since the time of Watt many
important innovations have been introduced into the steam engine
though the main features remain the same. The innovations were
needed in order to suit the circumstances of ever widening applica
tion of steam engines to various purposes. Watt always used steam
engines with low steam pressure., and of a static type. He was
evidently afraid of explosions. But engines using low pressure steam
are comparatively inefficient, as we shall see presently, and in modern
times high pressure engines have almost replaced the old Watt
engines necessitating the construction of special type of boilers.
Condensers in modern engines consist of a number of tubes
containing cold water kept in circulation by means of a pump, and
JT
212
HEAT ENGINES
CHAP.
Slide
Valve
Crank
Piston
Piston Rod
Gudgeon Pin
Wheel
are further provided with a pump to remove the air and water pro
& bv steam on condensation. m
Again in powerful engines the
Steam Ch©sT_ high pressure steam is not allowed
to expand completely in a single
cylinder. The steam is partly ex
panded in one cylinder and passed
on to one or more cylinders where
the expansion is completed. Such
engines are known as compound
engines and may consist of three
or four cylinders.
Richard Trevithick was. the first
to construct a 'locomotive 1 , i.e., a
steam engine which con Id draw
carriages on rails. He could
however, push his inventions to
financial success. It was left to
George and Robert Stephenson,
father and son, to construct the
First successful locomotive — the
"Rocket", and run the first rail
way train in 1829, between Liver
and Manchester. Robert
on was the first to apply the
steam ei Q ships in 181;:.
12. Efficiency of Engines and Indicator Diagrams. — The earlier
ventors of steam engine had no clear idea of the Nature of Heat,
ra rather than physicists, they did not make seri
. attempts at understanding the physi involved in the
running of a steam engine. They measured efficiency by finding out
the quantity of coal which had to be burnt per unit of time in order
to develop a certain power. Tin's was rather a commercial way of
measuring efficiei
An absolute measurement of efficiency is obtained from the first
law of thermodynamics. A heat engine is merely an apparatus for
rsion of heat to work. THe heat supplied is obtained by
finding out the calorific value of the fuel consumed by burning a
sample of the fuel in a bomb calorimeter (p. 63). If Q be the calori
fic value of the fuel consumed per unit of time and W the power
developed, we can define the economic efficiency t; as the ratio be
tween Q and W, viz. rj = W/JQ. Accordingly tj is the fraction of
the heat, converted to work.
For an ideal engine, ■>} should be unity. But actual experience
shows that i) is rather a small fraction. In Watt's days, it was only
h% ', nowadays even in the best type of steamengines, it hardly
exceeds 17%.
The question arises whether this lack of perfectness is to be
ascribed to the bad designing of heat engines, or whether there is
Shaft £
Eccentric
.—Alain parts i
;ine.
THE CARNOT ENGINE
213
ix.l
something in the very nature of things which prevents us from con
verting the whole amount of heat to work.
This question was pondered over by Sadi Carnot about a hur>
m ,,i years He showed that even with an ideal engine, it is
■possible to convert more than a certain percentage of heat to
work.
It is very convenient to represent the behaviour of an engine
by an indicator diagram and hence iu discussing the theory and per
formance of heat engines this is always
done. Suppose a certain amount of
gas is contained in a vessel at a certain
perature and pressure and occupies
a certain volume. Evidently the state
of the substance is uniquely represent 1>
ed by assigning its pressure and volume.
Thus we can represent the state of the
gas by a point A (Fig. 9) on a graph
such that the abscissa of the point re
presents the volume of the gas and the
i :ate represents the pressure.
Let the pressure and the volume
of the gas be changed to that corres
Ffe. 9. Ti:
ponding to the point B and suppose the pressure and the volume
i rnout this change are represented by points on the line All.
Then this operation is represented by the line AB on this dur
Such pv diagrams arc known as indicator diagrams*
\s proved on p, 4fi the work done by the gas in expanding against
a pressure p is p*S. In this case since p changes from point to point
ral work done by the gas in expanding from :•, to v* is equal
to f v * pdv and is evidently equal to the area AabB, The work is
taken to be positive if the diagram is traced in the clockwise direction.
It is thus clear that the indicator diagram directly indicates the
work done bv the engine during each cycle of operations, the work
being equal to the area a by the indicator diagram, indicator
diagrams are therefore of great use in engineering practice.
The indicator diagram gives the work performed by the piston
per stroke. Multiplying it by the number of strokes per second i.e.
bv twice the number of revolutions of the shaft, we get the power
indicated by the indicator which is generally expressed in horsepower
_;,.,. p a The power actually delivered by the engine
is measured by a brake dynamometer and is called the brake horse
wer, the ratio of this to the _ indicated horsepower is called the
medio:; in; I efficiency of the engine.
13. The Carnot Engine.— As we have seen, the function of the
steam engine is to convert the chemical energy stored in coal to
energy of motion by utilising the expansive power of steam.
214
HEAT ENGINES
[CHAP.
The machinery necessary for this purpose is, however, so com
plicated that one is apt to lose sight of the essential physical prin
ciples in the details of mechanical construction. Lee us, therefore,
discuss the physical principles involved in the running of a heat
engine. Three things are apparently necessary, viz., a source of
heat, a working substance, and machines. In the steam engine, the
source of heat is the furnace where heat is supplied by the burning
of coal. But we may get heat by a variety of other means, e.g. s
by burning oil, wood, naphtha, or even directly from the sun (solar
engines) , or from the inside of volcanoes (as is sometimes done in
Italy). We can, therefore, replace the furnace by the general term
"reservoir of heat/* For steam, we can use 'the general term
1 working substance," for any substance which expands on heating
can be used for driving heat engines. As a matter of fact, we have
got hot air engines in which air is heated by a gas burner or a
kerosene lamp, and pushes the piston up and down as steam would
do.
In addition to the three requisites mentioned above, we require
a fourth onc,_ viz., the possibility of having a temperature difference.
1 his at first^ is not so apparent, but can easily be made clear. In a
hot air engine, the heated air can push the piston outward since
the air outside is at a lower temperature. If there were no difference
of temperature, no difference of pressure could be created, hence the
machine would not work. We, therefore, requhe not only a sou
of heat, but also a sink, i.e., a heat, reservoir at a lower temperature
In .steam engines the surrounding air acts as the sink of heat or
con i
bserved that the fund the machine is to extract
a certain quantity of heat Q horn the heat reservoir F, convert a
part of it to work and transfer the rest to the heat sink G. He
also showed how these operations
should be carried out so that the
efficiency may be maximum.
Since whenever there is a differ
ence of temperature, there is a pos
sibility of converting heat to work,
the converse is also true, i.e. if we
allow heat to pass from F to G by
conduction, wc miss our opportunity'
of getting work. Hence we must
extract heat from F in such a way
that loss of heat by conduction is
reduced to a minimum. Carnot,
m« m ti  a 1 ^ lefore, thought of the following
s ' 10 :~V^. ld f ' £ arnot en «'» e ideal arrangement,
with indicator diagram. t 5
^ rr,. ,,s, * F T ' s a Jieat reservoir at tempera
ture T (Fig. 10) , G a heatsink at T f , S is the cylinder of the engine
containing a perfect gas instead of steam as the working substance
and fitted with a nonconducting piston. The walls 'of the cylinder
EL]
THE CARNOT ENGINE
215
are impervious to heat but the bottom is perfectly conducting. The
behaviour of the working gas is shown by the indicator diagram
showing the pressure and the volume of the gas at any instant. i>et
the following steps be performed :—
(1) Let the initial temperature of the gas within S be T and let
it be pitted in contact with F, and die piston moved forward slowly
Ah the piste, :  the temperature tends to fall, and heat will pa^
from F to S. The operation is performed very slowly, so that the
temperature ol the gas is always 7\ The. representative point on the
indicator diagram moves from A to B along an isothermal curve. 1 he
heat O cted in this process is equal to the work done by the
piston in free expansion, and is given by
= = f B pdv = RT log, £  area AabB. . . (1)
(2) F is then removed and H, which is simply a nonconducting
cap, is applied to the cylinder, and the piston allowed to move for
d (by inertia). Then the *as will describe the atflabatic BC and
will faU ill temperature. We stop at C when the temperature has
fallen to P. 'I he work done by the gas is given by
where piP — K"=£ & o v = AbV<
Since the pressure is now very much diminished, die gas has
lost its expansive power, hence in order to enable it to recover its
capacity Tor doing work it must be brought back to its original con
dition. To effect this we compress the gas m two stages: first,
isothermalh all ng the path CD, and then adiabaticaUy along DA.
The point D is obtained by drawing the isothermal T through U
and the adiabatic through A.
(B) During the isothermal compression, the cylinder is placed
in contact with the sink G at !P. The heat which i s developed
owing to compression will now pass to the sink. This is equal to
the work done on the gas and is equal to
= area CcdD. . (3)
. 0' = \°pd»=* tfT'log,^
(4) The cylinder S is now placed in contact with II and the
gas is compressed adiahatically. The work done on the gas by adia
batic compression is
Wi = f "/wfc = ~( T T')  area DdaA. .
It is thus seen that m z t=W4,
m
210
HEAT ENGINES
The net work done by the engine
W — w x \. w s w 4 — area A1JCD,
[CHAP.
(6)
The last result can be written down directly from the first law
of thermodynamics.
Since B and C Ik on the same ad i aba tic, we have by equation
24, p. 48., 1
■:'...■" ■'">■.  1
(£P" <?>
— I ■/• J — Pi the adiabatic expansion ratio.
or
Simi •
n
prf
or
= :r^= r .» the isothermal expansion ratio
have, therefore,
Q = RT log r, £' = i2:r log r
and Tr0/>' = i;.;;rT') log • r. .
r 3"' ""■•/:
Hence
afV'j
(S)
(9)
of the Cartiot en:
W T'
HIP < 12)
analysed the moi ion of the Camot cycle we now
I to .show that (I) it is reversible at cadi stage. (2) that no
engine can be more efficient than the Carnot engine, m that the
re of the working substance is immaterial.
14. Reversible and Irreversible Processes,— A process is
one which can be retraced in the opposite direction to that h
substance passes through exactly the same states in all t8 as
in the direct process. Further, the thermal and mechanical effects
at each stage should be exactly reversed, U., the amounts of heat
ived and of work done in each step should be the same as in the
direct process but with opposite sign. That is, where heat is
absorbed in the direct process it should be given out, in the reverse
cess and ; : , th and where work is done bv the working sub
stance m die direct process, an equal amount of work should be
DC]
RSIBLE AND IRREVERSIBLE PROCESSES
21?
done on the working substance in the reverse process. Processes in
which this does not take place are called irreversible.
For clarity we may add some examples of reversj
F 1 '■■••■' •  transfer of heat from one bodv to another can be
reversible only when the two bodies are at the same temperature.
In case of two bodies at different temperatures, the transfer of
heat occurring by conduction or radiation cannot be reversed and
the process is irn ible.
We shall now consider examples of reversible processes. It is
r from the above definition that the process of bringing an elastic
substance into a definite state of stress very slowly is revers
because for a given strain the substance has always a' definite stc
A convenient mechanical example of a reversible pro* afforded
by the performance of a spring balance in the following way; When
the spring is very slowly stretched work is done upon it. If, on the
other hand, it is allowed to contract slowly bv the same amount
the same amount of work is done by the spring, for die work done
in increasing the length of the spring by Si is equal to the product
of the force F and U. Both these quantities depend upon the si
of the spring at the instant and ave the same value whet]
l f c s P n ' : : ding or contracting. Thus the work done upon
n f.. be equal to the work done bv the ■
wlu  the reverse process and the process
[ f n ■'■• ■■'< however, that the stretching must be
d or reduced gradually by the c 'ion of a force
• mftnitesimaUy at every instant, from the stn
developed m the spri ,, part of the work will be spent
in setting up vibrations of the spring and this will
ability Such a process is called a quasistatic process and consists
essentially of a ,. m of equilibrium states.
The case of elastic fluids is analogous to that of the spring To
:v volume of the fluid there corresponds a definite stress
pressure, so that the amounts of work done during a balanced expj
sion or compression are equal. This is an important example o£ a
reversible process. It is important to note that the expansion should
be balanced otherwise whirls and eddies may he set up in the fluid
which will gradually subside on account of flui< i :i ion with the pro
duction of heal and thus a part of the mechanical work would be
lost. Such expansion or contraction may be either isothermal or
acLxababc and can be brought about easily bv applying pressure on
piston enclosing the fluid and adjusting the pressure to differ from
the fluid pressure by an infinitesimal amount.
Examples of irreversible processes are (I) sudden unbalanced
expansion of a gas, either isothermal or adiabatic, (2) TouIcThotri
expansion, (3) heat produced by friction, (4) heat generated when a
i ent flows through an electrical resistance, (5) 'exchange of heat
5? 1 "52! ! ? odles ,^ differ e nt temperatures by conduction or radiation
(n) diffusion of liquids or gases etc. Examples (1) and (2) exhibit
internal or external mechanical irreversibility, (4) and (5)' exhibit
thermal irreversibility, and (6) exhibits chemical irreversibility.
21 a
HEAT ENCINF5
[CHAP
A reversible process may be represented by a line on the indicator
diagram (p, v) out an irreversible transformation cannot be so
represented.
14. Reversibility of the Carnot Cycle*— It is now important to
notice that the Carnot cycle is reversible at each stage, i.e., instead
of abstracting the heat Q from a source T and transferring a part Q'
i sink T' and converting the balance Q  Q' to work, we can proceed
in such a way that the machine abstracts the heat <T from the sink
at T, then we perform the work W on the machine, and Q is trans
ferred to I source at 7\ This is done by proceeding along the
reverse route ADCBA, i.e., first allowing the gas to expand adiabati
cally from A to D, then allowing it to expand isothermally from D
to C in contact with the sink at V, the heat (T being extracted in
the process. Then we compress the gas adiabatically till we reach
the point B (temperature T) . Next S is placed in contact with the
source, the gas is further compressed isothermally till we reach A, and
heat (Ms transferred to the body at the higher temperature T.
The machine, therefore, acts as a refrigerator, :>,, by performing
the work W on it, we are depriving a colder body T* of the heat O .
An engine in which the working substance performs a reversible
cycle is called a reversible engine. Engines in which the cycle is
versible are called irreversible engines. Fur the Carnot cycle to
be reversible it is essential that the working substance should not
differ sei erature from that of the hot body and the
condensi l it is exchanging heat with them. There should be
of heat by conduction in the usual sense, for it. would be
[b] . This ires that the isothermal processes AR, CD
bed indefinitely slowly, and the source and the con
arge capacity for heat so as not to
change in temperature during the proot
ain the piston should move very slowly without friction. The
changes in volume should be brought about by very small changes in
the load on the piston in both the .isothermal and adiabatic processes,
so that the difference between the external pressure and the pressure
of the gas should always be infinitesimally small. Tims the Carnot
cycle postulates the existence of stationary states of equilibrium
while in an actual process the physical state is always changing.
Further there should be no loss of heat by conduction from the
to the piston and cylinder. Tt will thus be seen that the Carnot cycle
with its perfect reversibility is only ideal and cannot be realised in
practice,
Nevertheless, it should be noted that for theoretical purposes die
deviations from the ideal state may be lightly neglected if they are
not an essential feature of the process and if they can be diminished
much as may be desired by suitable devices and then corrected for.
16, Carnofs Theorem.— The idea of reversibility is of the great
est importance in thermodynamics for the reason that, working
between the same initial and final temperatures, no engine can I
IX.]
CARNOT S THEOREM
219
■ent than a reversible engine. This is known as Garnet's theorem.
We now proceed to prove this important theorem,
Suppose we have two engines R and S, of which R is a reversible
engine and S an irreversible engine. If possible, let S be n
efficient than "R, Suppose S absorbs the heat £) from A, converts
l W f to work and returns the rest, viz., Q~W to the condenser.
Let S be coupled to R (Fig. 11) and be used to drive R backwards.
We are using R as a refrigerator. It thus abstracts a certain
amount of heat from B, has the
work W performed on it, and
returns the same heat Q to the
source A. The amount "of heat
that R abstracts from B must
equal QW* Now since S is
assumed to be more efficient than
R, W>W and hence Q— FF>
Q  W r , vi&, R abstracts more
heat Iron: B than S restores to
it. Thus the net result is that the
compound engine RS abstracts
heat (WW) per cycle from E
and converts the whole of it to
work,f while the source is un
N
/*
Hot Bod 1 ?
A
>
v R ^
ill
•..,
y
B
QW'
\
• II,— Coupling a reversible and
an irreversible engine.
affected. We are thus enabled, by a set of machines, to deprive a
body continuously of its heatcontent and convert the whole of it
work without producing any change in other bodies. The machine
would tii us work simultaneously as a motor and a refrigerator and
would be the most advantageous in die world. It does not violate
the first law for we are creating energy out of heat.
Impossibility of perpetual motion of the second type. But still
the process is quite as good as perpetual motion! of "the first kind.
for heat, is available to us in unlimited amount in' the atmosphere, in
the soil or the ocean, and if the process were feasible, it would give
us all essential advantages of a perpetual motion machine, viz., that
of getting work without any expenditure, though not without enere
Human experience forbids us to accept such a conclusion^ Hence
we conclude that no engine can be more efficient than a reversible
engine.
Again, if we assume that a reversible engine using a particular
* Reproduced from Ewing's Steam Engine by the kind permission of Messrs
Maemil3a.ni & Co.
t X!l' £ work ^' ~ ^ which is available, mav he used in driving a motor.
J this was a term in use amongst the medieval philosophers who. thought that
a machine might he invented some day which will create work out of nothing
for some mechanical contrivances which , v cre intended to produce perpetual
motion see Amlade, fiuw, 3 «gc 14. The gradual evolution of the Law of
ration of Energy lead; : to the first law of thermodynamics showed that
is purely chimerical, since energy can never be created out of nothing
;'," V 11 ''' ^ transformed from one form to the other.
1 1 Tins result of human experience is the fundamental basis of the Second
Law of Thermodynamics. In fact, Kelvin stated the second Jaw in this form
t*or fuller discussion, set Chap. X.
220
HEAT liNGINE*
[CHAP,
working substance is more efficient than another working with a differ
,1,1 substance, we arrive by a similar argument at the aame absurd
result. Hence the efficiency is the same for all reversible engines
this is the highest limit ; efficiency of any engine that can
be constructed or imagined. This is Camot's theorem. Thus rever
sibility is the criterion of perfection in a heatengine. Hence we sec
that the efficiency of a reversible engine is maximum and is lnde
pendent of the nature of the ing substance. We chose perfect
as our working substance, since its equation of state being known,
we can easily evaluate 17.
Note : — It may be mentioned that Carnot was ignorant of the
true nature of heat at the time when he published his theorem of
efficiency He followed the old caloric theory in his speculations
according to which the quantity of caloric O contained in a gubsta
was invariable. It was Clapeyron who showed that Camot's argu
ments and result* remained intact when the Kinetic Theory of Heat
was introduced.
17, Rankice's Cycle. — In the steam engine, as we saw in the
foregoing pages, the working substance is a mixture of water arid water
vapour. ° If, 'however, we perform the same Carnot cycle with this
fluid as with perfect gas the efficiency would also be the same {Camot's
theorem) . But the Carnot cycle was performed with the working sub
stance always in the same cylinder. We have already seen that with
a xi vent, the je of heat consequent on alternate heal
and of the cylinder the modem engine; are provided v.
Still, howevi work
ice were to perfon e the efficiency wouldbe
with the organs so separated the adiabatic corny
working substance in the last stride of the cycle becomes
imp] : e . Hence the cycle is modified into what is known as
 cycle.
The Rankine cycle is represented in Fig. 12. A'B represents the
conversion of water into steam in the boiler at temperature F 3 and
e pi and its admission into the cylinder, EC the adiabatic
"on in the cylinder, CD the transfer of steam from the cylinder
t:r at T 3 and pa and its condensation, and DA the
transfer by a separate feedpump to the boiler. This separate feed
pump transfers the condensed water at D
at T 2 and p s to the boiler and the pressure
is consequently raised From p% to p x . At
A the water is heated from T 2 to T 2 in
the boiler and the cycle begins afresh.
The indicator diagram for an engine per
forming the ideal Rankine cyi repre
sented by ABCD and this also represents
the work done by the engine. The area
Fig. 1 The Rankine cycle. FADE lily called the Jecdpuinp term,
is extremely small and is generally
lected. The work done by the engine may, therefore, " be put equal
IX,]
TOTAL HEAT OF STEAM
221
to the area. FBCE. This area can be calculated in much the same
way as on p. 215 if we know the equation of state of steam. This
is cumbrous and in engineering practice a simple procedure is adopted.
Let us introduce the total heat function H = U ■{• pV. Then
dll = dU + pdV I Vdp — Vdp
for an adiabatic process, since in this case (Chap. II, §23)
\.pdv=a.
Integrating between the limits B and C (Fig. 12) we get
ighthand side represents the area FBCE. Hence the work done
in Rankine's cycle per grammolecule of steam is approximately equal
to the heatdrop H n —He This will hold whether the steam is super
heated, saturated or wet.
The heat taken in by the working substance is that required to
convert water at p y and T z into steam at p t and TV This is equal to
H B  [H D  (/>i •  p^ V w \H n II n approx.
where 11%, Hn denote the total heat of steam at pi, T } and of water
at y. , lively, V w the volume of v*»ter at D. The term
(pi — pi) V, v 1 nts the area FADE. The efficiency is
Hs — Hq
The values of the total heat function are readily obtained from steam
tables or charts which have been prepared for engineering work as
explained in § IS.
It will be v seen that the efficiency in Rankine's cycle is less than
in Carnot cycle for in the former some heat (viz., that required to heat
the feedwater in the boiler from T 2 to T 7 , is taken at a lower tem
perature. In actual steam engines the efficiency is only about
60 — 70% of the Rankine ideal and their indicator diagram resembles
Fig. 12 with the corners rounded off.
18, Total heat of steam. — For calculating numerical values* it
is usual to assume that the total heat or enthalpy of saturated water
. water in equilibrium, with i' : ; saturated vapour) at 0*C is ;••: .
and calculate the changes in total heat. Then the total heat of steam
at (9°C is defined as the amount of heat required to convert 1 gram
of water at 0°C into steam at &°C Regnault gave for the total heat
of steam H the following empirical formula
H = 606,5 + 0.305 $
where denotes the boiling point. The total heat may be computed
as follows :
Let us raise the temperature of 1 gram of water from 0°C to its
* Si.::' Keesom and
data on ."• ■ .' cmd solid
■X 1936).
ICeyes, Thermodynamic Properties of Steam, inch
lolid phase, (Wiley, 1936), or Calletiiiar, Steam
222
HEAT F.NGTNES
[CUA!\
boiling point (t°C) , Denoting the average specific heat at constani
pressure by c, the heat required for this process will be U if we
neglect* the small work done due to pressure changes. Thus the total
of water at t°C is H. Now let us evaporate this liquid at its
boiling point t°. Consider the state of affairs when q gram of vapour
has been produced leaving (1q) gram yet to be evaporated. Then
q is called the dryness fraction. The volume of. vapour produced is
qv and the increase in volume is q(viv) where v is the specific
volume of vapour at t°C and w the specific volume of water. Hence
the external work done is pq (vw) in mechanical units. Therefore
the gain in total heat during evaporation is
qL— qi \ pq (v  to)
where * h the increase in internal energy when unit muss of the liquid
is evaporated. Hence the total heat of steam at dryness fraction q i
H=ct\ qL.
For dry saturated steam (q =. I), this becomes
B = St + L.
If the dry saturated steam at t"C is further heated, its temperature
rises and we now speak of it as superheated vapour. In the case of
am the superheating is usually produced under such conditions that
can assume that the superheating is carried out at constant pres
u i the temperature of the superheated vapour to be t t °C
that there are (ti — i) degrees of superheat. If c/ he the mean
ilir heat n\' the vapour at constant pressure, rhe total heat of super
H s ~a{L+c p '(t x 
INTERNAL COMBUSTION ENGINESf
19. Historical Introduction.— Like other types of heat engines
the this type also dates front medieval limes, Ch. Huygens.
* M ore rigorously
(ill — d (« + h) — <*« + Pdv f
and
(T<
.'t>
:QS\
v H
= Tds + vdp.
+ Xr
dH =
c
dH
dT
)."+(X\
*i;^fiMfr)>
and we must integrate the righthand side from 0"C (To'K) to the boiling point
T and the pressure from the saturated vapour pressure ^,> at To to the saturated
vapour pressure p at T,
t These are called internal combustion engines because in them heat is. pro
by combustion of fuel inside the cylinder in contrast with the steam engine
which may be called external combustion engine I cans h this tribe heat is
produced in the boiler. The interna] combustion engine is much more efficient
than the steam engine because the working substance can he heated to a much
higher temperature (20OO°C).
IX.]
THE OTTO CYCLE
%>$
the great Hutch physicist, proposed in 1680 an engine consisting oi
a vertical cylinder and piston, in which the piston would be thrown
upwards by the explosion ot a charge oL : gunpowder. This would I il
the cylinder with hot gases which would eventually cool, and
pis: Id be forced down by gravity. Each stroke would require
a fresh charge of gunpowder.
This engine was never used, probably owing to the difficulty oi
introducing fuel alter every explosion but the idea pcrsisusd. The
discovery o£ combustible gases (coal gas, producer gas...) and of mine
ral oils brought the idea within the range of practical possibilities, as
the difficulty of supplying fuel promised solution. But many long:
years of practical and theoretical study were necessary before such
engines became commercially possible.
7t is not possible to go into
the details of these early at
tempts, and it will suffice to
describe and explain the action
of the two types* which have
survived, viz., (I) the Otto
engine in which heat is absorb
ed ai S) the
'•in whij li heat is
bed al constant pressure.
ly BO per cent of" internal
combustion engines today are
of the Oi to type. We shall com
pare the action of these engine'
with that of the ideal Carnot
engine in which heat is absorb
al constant temperature.
20. The Otto Cycle,— The
Ottof cycle we are now going
Lo describe was originally pro
d by Beau de Rochas
.1 ., but the practical diffi
culties were first overcome by
L876. The working of
iic. will be clear from
I The engine consists of the cylinder and the piston, the
cyl ider being provided with inlet valves for air and gas (if gas is
. lor combustion) and exhaust, valves. The opening and closing
ese valves are controlled by the motion of the piston. There
our strokes in. a complete cycle:
♦Sometimes internal combustion engines are divided: into (1) gas engine,
(2) oil and petrol engines, ft is, however, more scientific to divide them with
regard I they follow and not with regard to fuel. ' Hence '
have been divided into the constant volume, constant pressure and constant
raturc type,
t Nikolaus Oi I8M,), born at Schlangenbad in Germany, 3s best k:
;:.. the inventor of the fourstroke gas engine.
Fig. 13. The four strokes of the Otto
• ;.iue.
224 HEAT ENGINES [CHAP.
(i) The Charging Stroke. — In this the inlet valves are open, and
a suitable mixture of air and gas is sucked into the . the
forward m i of the piston.
(if) The Compression Stroke^ During this stroke all the val
are closed, and the combustible mixture is compressed adiabatics
to about l/5ih of its original volume by the backward motion of I e
ton. The temperature of the mixture is thereby raised to about
; a
At the end of the compression stroke, the mixture is fired by a
series o£ sparks.
(iii) The Working Stroke, Hie piston is now thrown forward
witl oree, since owing to combustion a large amount of beat
developed which raises the temperature of the gas to Lboul 2000 D C
and a corresponding high pressure is de\
(iv) The Scavenging or the Exhaust Stroke,— At the end of the
third'. stroke, the cylinder is filled with a mixture of gases which is
useless ; iher work. The exhaust valves are then opened, and
die piston moves backward and forces the mixture taut.
scav is complete, a fresh charge of gas and air is sucked in
and a fresh cycle begins.
The tli uamical behaviour is illustrated in the indicator
■diagram (Fig. 1
It should "■ inhered that air is the working substance in
Otto engine, the function of gas or petrol being merely to heat
the air by its combustion. EC represents
the suction stroke (gas and air being sucked
at atmospheric pressure) . CD represents
the compression stroke. At D (pre
about 5 atmospheres, temperature about
G00 C C) the mixture is fired by a spark. A
large amount of heat is liberated, and the
representative point shifts to A (tempera
ture 2000 a C, pressure about 15 atmos
pheres), volume remaining constant, AB
sents the working stroke. At B the
ust valve is c and the pre;
[alls to the atmospheric pressure GV t .
is the scavenging stroke.
The efficiency of the engine can be ea
if'ed out. The amount of heat added
b C a (T a —Tj) where C 9 is the : heat
at constant volume. (We suppose that this quantity retains the
same etween 6GQ°C, and 2000° C. which is only approximately
true.)
Heat rejected = C s ( T b  T r i .
Hence efficiency tj=1=t ,J' . • . (1$)
Im.lt. 14 I
Otto cycle.
t
IX.]
thk orro CYCLE
225
From the relation TV** = constant for an ad i aba tic (see p. 48,
equation (24), we obtain
T* iVt\* 1 T d
(14)
where p is the adiabatic expansion ratio. Hence
r.r.np
T B T*
Therefore the efficiency
='(1)
I ',>'
 (15)
We obtain a similar expression for efficiency in terms of the adia
compression ratio as we get m a Carnot cycle (see equation 12,
p. 216). The question arises : why not try to make the Carnot cycle
a practical possibility? '
We shall now treat this question in more detail. In desiei
a machine, other factors have to be taken into consideration to
Lion of theoretical efficiency. The engine must be, in
the first, place, quickacting, i.e., a cycle' ought to be completed as
tunckly as possible. This practically rules out the Cajnot engine as
Les up heat at constant temperature, the process is therefore
and the engine becomes very bulky and heavy in relation
ait. In the Otto engine, the absorption of heat is
almost instantaneous. But even where quickness is not the deciding
factor, there are other weighty arguments against the adoption of the
Larnot cycle. Pressure inside the cylinder varies during a cycle and
Lie machine must be so designed that it can withstand the maximum
pressure. Hence practical considerations impose a second condition
t.e the maximum pressure developed inside the cylinder durjW a
cycle should not be too great. Further, we must have a reasonable
amount of work per cycle.
Now due to the adiabatic compression in the last stage of the
Carnot cycle an enormous pressure is developed. A detailed consi
deration taking numerical values shows that working between the
same two temperatures, say 2040°K and MCPK the Carnot engine
ops a maximum pressure of about 1000 arm., while in the Otto
engine it ; « only about 21 atm„ though of course, the efficiency is
reduced from 83% to about 44%, The Carnot engine also rec
a large volume for the cylinder. These considerations show clearly
tAat Che Carnot engine is quite impracticable. It will have to be
very bulky and very stout and the power output will be extremely
spall compared to its bulk.
In the case of the Diesel cycle which is described in the next
section the maximum pressure developed is about 35 atm and the
efficiency rises to about 55%, It is for this reason that the Diesel
engine is employed in cases where we want a large output of work.
15
mi
HEAT ENGINES
[CHAP
DIESEL CYCLE
227
hnl KbMm
D J A
21. Diesel Cycle. — Diesel* was dissatisfied with the low efficiency
of the Otto engine and began investigations with the idea that
efficiency of the Carnot cycle may be reached by certain other contri
vances. ' He did not succeed in his attempt, but was Led to the
invention o£ another engine which, for certai rposes, presents
ked advantages over' the Otto engine,
>w since
it is easily seen that if the com
pression ratio p can be still further
pushed up f ^ would substantially
increase. But in the Otto ei
p cannot be increased beyond a
certain value (about 5) otherwise
the mixture would be fired during
pression before the spark
passes.
In the Diesel engine no gas or
petrol is introduced during com
pression. The indicator diagram
tor the cycle is shown in Fig. 15.
In the first stage pure air is sucked
in (EC in Fig.) an* I I to ah nut one seventeenth of its volume
in fig.) . A valve is then opened, and oil or vapour is forced under
IS.— The Diesel •
Fig. 16.— Strokes in a Diesel Engine.
(o) Beginning of suction stroke; airvalve open,
i :'.',' i T: entitling of compression stroke; all valves closed.
(c) BcRintiing of working stroke; oil valve open.
(d) Working stroke in progress. Etui at End injection ; all valves closed,
(j?) Beginning of scavenging stroke; exhaust valve open.
♦Rudolf Diesel (18531913), barn in Paris of Get man parentage was the
inventor of the heavy oil engine. He was engineer at Munich.
ure. The oil burns spontaneously as the temperature in the
is about 1000°C. and above the ignition point of the fueL
>f oil is so regulated that during combustion, as the
ptsinh forward, Lhe pressure remains constant (DA).
. when the temperature has reached the maximum value,
e supply of oil is cut off. The piston is allowed to
', describing die adiabatic AB. At li a valve is opened,
mill pressure drops to C. CE is the scavenging stroke in which the
fixture is forced out, and the apparatus becomes ready
ycle.
cycle can be performed in a cylinder of the type shown
16. The cylinder is provided with the air inlet, the oil supply
md die exhaust valves. The action of the cycle will be very clearly
folio wi ' 1 1 from the figure,
instead of opening a valve at B we may allow the gas to expand
F where the inside pressure has fallen to atmospheric pres
.iiii', and Lhe air mixed with the burnt gas be forced out by a back
FE. As this would involve a rather large volume for the
, this procedure is not adopted in actual practice.
all now calculate the efficiency of the Diesel engine. Let
m i"i calculate the efficiency for the imaginary cycle AF6D,
Heat taken = C P ( T a — T d ).
Heat rejected = C p {T f T t ).
relation p = cT v! ^ 1 ' 1 (equation 25, p. 48) we have
<*«* T d T a T m T d
m
ii
T f T c
f a f d
(yl)!y
T f T c
(£:)
vi
(16)
(17)
e for the efficiency' the same formula as for the Carnot
tto cycle. But in the Diesel engine the cylinder is designed
m pressure of about 34 atmospheres. This determines
1 1 on ratio because p\fp t ■ p y =34. This gives p=12.6
which is a substantial improvement on the OltO engine,
ted above, we have to take the cycle DABG. Heat
constant volume. Hence heat rejected is equal to
c v (T b r c ).
" v " ~C fi (T A T d )
Id h wer value of about 55$>.
(18)
HEAT ENGINES
CHAP.
The Diesel engine, therefore, consumes less End than the Otto
engine, bun els it has to withstand a higher pressure it must be more
robust. Also the mechanical difficulties are much greater, hut they
have been successfully overcoi
22. SeiniDieset Engines.— We have seen that in Diesel engines
To avoid these die SemiDiesel engine, also called the hotbulb <?.'
has been invented, In this air is compressed to about 15 to 20 aim.
instead o£ 35 atm. This air does not yet become hot enough to ignite
the oil and is consequently passed through a hot bulb where it is
heated by contact with the bulb to the required temperature. The
hot bulb 'is simply a portion of the cylinder which is not cooled by the
water jacket. Tt consequently becomes much heated and serves to
heat the compressed air. Next the oil (fuel) is injected into this hot
bulb where, by the combined action o£ the compressed air and the hot
bulb, it ignites. For starting the machine the hot bulb must be
heated by a separate flame. After a few cycles the bulb becomes
hot and begins to function properly. Then the external flame is
removed. The semiDiesel engines nowadays are generally two
stroke oil engines without automatic control valv
23. The "National" Gas Engine,— We shall now describe a
typical Otto engine. Fig. 17 shows a 100 horsepower "National" gas
: ne.
Fig. 17.— The "National" gas engine.
The engine is a stationary horizontal one having a single cylinder
A A. The piston PP is attached to the crank shaft by the connecting
rod CC and (its gastight into the cylinder AA. H is the combustion
chamber having the inlet valve 1, the gas valve G and the exhaust
valve E. The cylinder, the combustion chamber and the valve casings
are cooled bv the water jacket WW. L is the lubricator, V, V are
rx,]
STEAM TURBINES
229
plugs which may be removed in order to examine the valves. M is
the ignition plug where a spark is produced by the "Magneto" and
serves to ignite the charge. The various valves and the magneto are
worked by suitable earns mounted on a shaft driven by the crank
shaft.
The working of the machine is identical with that indicated in
Fig, 13. The machine is started by rotating the flywheel so
the piston commences its outward stroke. An explosive mixture is
introduced and ignited and the piston is thrown forward. Then the
working follows as indicated on p. 223.
24. Diesel FourStroke Engines.—  These engines employ a cylin
der and valves of the type shown in Fig. 16* In constructional de
sign they are not much different from the Otto engine. They are
further provided with a high pressure air blast for injecting the liquid
fuel.
25. The Fuel*— Some engines employ gaseous fuels the chief of
which are coal gas, producer gas, cokeoven gas, blastfurnace gas,
water gas and natural gas (chiefly methane) . The common liquid
fuels are petrol, kerosene oil, crude oil, benzol and alcohol. Tn the
Otto i tigine, if liquid fuel is used, it must be first vaporised and then
s generally employ crude oils and hence their
fuel ' heap.
26. Applications. — During the brief period of fifty years the
internal combustion engines have been much developed and employed
for a variety of purposes. The steam engine, we have seen, is very
wasteful of fuel, but the internal combustion engine is much more
economical and has consequently been widely employed.
The Otto fourstroke engine is usually employed in motor cms
and aeroplanes. Petrol vapour is used as fuel in both cases. Four
stroke Diesel engines are largely employed in driving ships. Where
very large power is necessary, the twostroke Diesels ate in common
use. Diesel engine is more efficient and more economical of fuel.
Still has combined a steam engine with a Diesel engine and has
obtained much greater efficiency in steam locomotives (SLillKitson
locomotives/'!
STEAM TURBINES
27. The Steam Tnrblne. — We shall now give a brief description
of the engines belonging to the windmill type. Windmills are set in
motion by the impact of wind on the vanes of the mill, and at first
sight it may appear strange why they would be classed as heat
engines. But a little reflection will show that the wind itself is due
to the unequal heating of different parts of the earth's surface, hence
the designation is justified. Etit windpower is rather unreliable,
hence steam turbines were invented in which the motive power is
supplied by the artificial wind caused by high pressure steam. There
two types of turbines :_ — (1) The Impulse Turbine in which steam
issuing with great velocity from properly designed nozzles strikes
HEAT ENGINES
CHAP,
against the blades set on a turbine wheel and in passing through them
has its velocity altered in direction. This gives an impulse to the
Wheel which is thereby set in rotation. (2) The Reaction Turbine
is somewhat similar in principle to the Barker's Mill which, as a
laboratory toy, has been known to generations of students. In this
the wheel is set in rotation in the opposite direction by the reaction
produced by a jet of steam issuing out of the blades set in the wheel.
Though the steam turbines are simple in theory, in practice great
mechanical and constructional difficulties had to be encountered, but
thanks largely to the efforts of De Laval, Curtis and most of all to
Sir Charles Parsons who may be described as the James Watt of the
steam turbine, these difficulties were successfully overcome. Today
the turbine is a prime mover of great importance and is aim
exdusivcly used as the source of power for all large power land
stations for the generation of electricity and propulsion of big ships.
Their great advantage lies in the fact that instead of the reciprocating
motion of the ordinary steam engine, a uniform rotary motion is
produced by a constant torque applied directly to the shaft. They are
more efficient than other engines because there is no periodic change
of temperature in any of their parts and hence no heat is wasted.
The full expansive power of steam can be utilised under suitable con
ditions.
28. The Theory of Steam Jet*.— The essential feature of the
turbines is that, in them the heat energy of steam is first employed
itself in motion inside the fixed nozzles or bladings
(in reaction turbines) and this kinetic enej
is utilised in doing work on the blades. The
fixed nozzles are so designed as to convert
most efficiently the thermal energy of steam
into its kinetic energy and produce a con
stant jet. We shall, therefore, first study the
formation of jet under these conditions. We
shall derive an expression for the kinetic
energy imparted to the steam in traversing
the fixed nozzle from the region of high
pressure B (Fig. 18) to the region of low pressure C.
Let the pressure, volume, internal energy and velocity of steam
in E per gram be denoted by p lt v lt ih and w ± and the correspond ins;
i C by fa v., Us and w z . Then the gain in kinetic energy
of steam per gram on traversing the nozzle is (u> 2 2 n> x s )/2j cah
The net work done on steam in this process as on p. 138 is p^> : — p$v s
while the loss in internal energy is u t u^. Thus on th nption
that the process is adiabatic we have from the first law of thermo
dynamics,
rjj {w^w^) = p t Vj; aa^Aj— k tt , . (19)
i.e., the gain in kinetic energy is equal to the heat drop. Generally
the initial velocity is sensibly zero and hence die kinetic energy of
the issuing jet is given by the heat drop hi  h 2 .
Fist. 18— The fixed
IX.]
THE DE LAVAL TURBINE
281
£^~ m^E3*
If we now assume that there is no loss of heat energy through
friction or eddv currents in the nozzle, the heat drop At feu equal
to the heat drop from the pressure fa to p & under isen tropic condi
tions. This can be readilv obtained from steam tables and represents
under most favourable condition the maximum work obtainable from
the turbine. Under ideal conditions the ordinary steam engine would
also yield the same amount of work, i.e. } equal to the heat drop. The
neater efficiency of steam turbines is, however, mainly due to the
following two causes:— (1) in the turbine diere are nowhere any
periodic changes in temperature as are found in the cylinder of the
steam engine ; (2) the turbine is capable of utilising lowpressure
steam with full advantage.
29. The De Laval Turbine.— The velocity which is theoretic!
attainable in a steam jet is, therefore, enormous, and this causes
extraordinary mechanical difficulties.
For if we want to utilise the whole of
this energy., by allowing the jet of
steam to impinge on the blades or
buckets of a turbine wheel, simple
math: ! considerations show that
the velocity of the blades should not
be short of' onehalf the velocity ol the
or maximum efficiency we
should have a speed of the order of
100 metres/sec. at the periphery of the
wheel which carries the blades. The
number of revolutions per minute ;
sary to produce such peripheral speeds
for a wheel of moderate dimensions
reaches the value of about 30,000 and
such speeds are impracticable for two
reasons. First, th no construc
tional materials which can withstand
such enormous peripheral speeds.
The rotating parts, would fly to pieces on account of the large
centrifugal forces developed and there would be mechanical vibration
of the rotating shaft. Secondly, if the shaft is to be coupled to
another machine which should usually rotate at a much, lower speed,
say 2000 r.p.m., the problems of gearing become almost insuper
able. These difficulties were first successfully overcome by De Laval
for lowpower turbines. For the gearing down, he used doublchel
! • wheels with teeth of specially fine pitch. To enable the requisite
peripheral speeds to be obtained' with safety, the turbine wheel was
made very thick in the neighbourhood of the axis (see Fig. 19) so that
it can withstand the stresses set up by rotation and further tile shaft
made thin.
A De Laval turbine is illustrated in Fig. 20. The steam jet
issuing from the fixed nozzles impinges on the blades, has its direction
Fig. 19.— Vertical section
wheel of De Laval turbine.
232
HEAT ENGINES
CHAP.
changed by them and thereafter escapes to the exhaust. Due to this
action of the steam the
blades are set in rota
tion. These are known
as 'impulse turbines' be
cause the steam in pass
ing through them is not
accelerated but has only
its direction changed.
The interaction of steam
with the blades is clear ly
indicated in Fig. 21. In
these turbines the bli
are 'parallel*, i.e., a
passage between two
blades has nearly the
same crosssect
everywhere so that the
blades offer very little
obstruction to the steam
jet and hence the p
sure of the steam re
mains practically the
same on both sides • of
the wheel.
Fig. 20. A De Laval ti
30. Division of tfie Pressure Drop into Stages. Rateau and
Zoily Turbines. — As a prime mover, the De Laval turbine is quite
efficient for producing
genera
ed steam must be
used so. that the j
sure drop in the
would be enormous, and
ice for maximum effi
ciency the blades must
move with very high
velocity. .Such large
velocities are incompati
ble with safety. For this
reason, the entire pressure drop is divided into a number of stages.
Each stage consists of one set of nozzles and one ring of moving
blades mounted on a wheel fixed to the shaft. Across each 
the pressure drop is small so that the blade velocity necessary for
maximum efficiency is considerably reduced and can be easily attained
in practice. This principle of pressure compounding was first given
by Parsons and later adopted by Curtis, Rateau and Zolly to the De
Laval type. Rateau turbine has 20 to 30 stages. Zolly has about 10.
l r !.
21, — Action of steatn on the blade*
a De Laval turbine.
IX.]
REACTION TUUB1N£S
m
Steam
31. Reaction Turbines— Parsons' Work.— Parsons* began his re
searches on steam turbine about the same time as De Laval (1884) »
but he did not accept the fetter's solution of the problem, and
went on with his own investigations which were completed by
about 1897. His researches resulted in the evolution of a completely
; i — the socalled Reaction Turbines. The mechanism of this
type is shown in Fig. 22*
The essential feature of the Parsons' turbine is its blading
system. This consists of alternate rings of a set of fixed blades
mounted on the inside o£ the cylindrical ease or the stator, with a
set of moving blades mounted on the shaft or the rotor, arranged
between each pair of rings of fixed blades as shown in Fig. 5
There are no separate nozzles
for producing a jet, but the
fixed blades ace as nozzles. It
will be noticed that the blades
have convergent passages, i.e.,
the crosssection of the passage
goes on decreasing towards die
exit side.
the s t e a m passes
through the moving' blades,
is a drop of pressure in
equence of which the
steam jet increases in velocity
as it traverses the blade
age, and as it issues out, a
backward reaction is produced
on the blades which conse
Hill
Fdhk
Moving
Btadea
"Fixed
Blades
Moving
Bkdes
JJJJJ>
JJJJJJ
Fifi. 22.
Tbe blading _ system
M turbine.
of
quently begin to revolve. The
steam then "again passes to the next set. of fixed blades where it again
acquires velocity and enters the next set of moving blades nearly
perpendicularly.
The moving blades are secured in grooves on the rotor shaft
with their lengths radially outwards and parallel to each other. The
fixed blades are secured' in grooves in the enclosing cylinder and
project inwards almost touching the surface of the spindle. The
steam is admitted parallel to the axis of the shaft (axial flow type) .
One pair of rings of moving and fixed blades is said to form a stage,
and .generally a turbine has a large number oE stages.
Pressure Compounding. — On account of die peculiar form of the
bladepassages, the pressure goes on falling rather slowly, so that
hole pressure drop is spread over a large number of stages which
may amount to 45 or even 100. In consequence of this arrangement
the velocity of the jet is very small compared to that of the De
Laval type and the requisite blade speed is 100 to 300 ft. per sec.
* Charles Algernon Parsons (18541931) built the fir<L reaction turbine in 1886
at NewcaslleonTyne. He was awarded the Copley & the Rumford medals ot
the Royal Society for hh invwitions.
234
HEAT ENGINES
CHAP,
This reduction of velocity brought about by the distribution of avail
able pressure over a large number of stages, constituted the master
invention of Parsons which was also applied to other turbines with
great advantage as already mentioned.
We have stated above that the steam enters the moving blades
perpendicularly. _ In actual practice, however, the steam enters the
moving blades with some relative velocity giving it an impulse as in
De Laval's turbine and so the turbine works partly by impulse and
partly by reaction. Thus it will he seen that the distinction between
'impulse' and 'reaction' turbines is more popular than scientific and
should not be taken too literally. All turbines are driven by 'reac
tion/ due to the alteration of velocity of the steam jet in magnitude
and direction. The real distinction between the two types is that in
the 'impulse' turbines the reaction is due to the steam jet being
slowed down while in the pure 'reaction' turbine the reaction is pro
duced by the acceleration imparted to the steam in the g blades
themselves on account of the pressure drop. Still, however, by
simply looking at the blades one can say whether the turbii
the 'impulse* or 'reaction' type, the impulse blading being charaete
rised by parallel passages and the reaction blading by the convergent
passages.
During the last thirty years highpressure ga . i: been uti
Used in place of steam to drive the blades of a turbine. In the inter.
nal combustion gas turbine the combustion of some oil serves to heat
sed air to a high temperature and high pressure, arid this
high pressure gas drives the turbine wheel.
32. Alternative types of Engines. — During the Second World
War Chi developed in Germany and Italy
mainly with the idea of devising a powerful means of propulsion for
>speed aircraft. In a modern turbojet a mixture of air and fuel
ticked in and compressed by a compressor and then ignited so as
are. rhis high pressure gas traverses a ti
bine where it gets accelerated and the pressure falls, the gas escap
ing at the rear o[ the machine with a high jet velocity, The machine
on account of the reaction thus furnishedj move yard with tre
mendous velocity. In this way the modern jet plane has been able
to attain supersonic velocities le< velocities greater than the velocity
of sound.
Mention may be made of the Rocket and the guided missile
whose use by the Germans in World War II as V2 rocket caught
world imagination and whose potentialities for space travel are bou
to make it an important tool of the future for scientific research. The
essential difference between a rocket and a jet aft is that the
Cornier carries its own oxygen supply and thus can operate in vacuum
while the latter obtains oxygen from the air. The propellant. solid
or liquid, ignites producing a jet and the rocket takes off. The
final rocket velocity V r — zv log f M* where n> is tlje jet velocity
and M* denotes the ratio of the initial mass of the rocket to the final
mass after combustion. In the twostage rocket a smaller second
IX.
TI HEMODYNAMICS OP REFRIGERATION
235
rocket is placed hi the nose which ignites a lew minutes after take
off and the second rocket detaches itself and forges ahead under its
own power with a tremendous velocity. Such a multistag
was used by the Russians to launch the first artificial earth sal
on 4th October, 1957 and the first artificial planet on 2nd January,
1959 The thrust developed by these rockets is enormous.
THERMODYNAMICS OF REFRIGERATION
33, Tn Chapter VI we described several refrigerating machines
and considered the general principles upon which their action depends.
We shall now investigate the problem of die efficiency of these
machines. In order to compare the efficiency o£ the various machines
we shall introduce a quantity called the coefficient of pe: nee of
a refrigerating machine. If by the expenditure of work W a
is deprived of its heat by an amount q, its coefficient of performance
is given by q/W,
Tn Section 15 we saw that if a Carnot engine with a perfect gas
as the working substance is worked backwards between the source
of heat at T] and the condenser at T.,, the working substance abs
tracts an amount of heat (X from the condenser and yields (h
to the source where Q^ and Q* are connected by the relation
Qt/Ti = Qg/Tjf The work done in driving the engine is Ql — Qz
Hence the coefficient of performance of a Carnot engine working
refrigerating machine is
a.
r.
di  Qa " **  T f '
(20)
It can now be easily proved much in the same way as on p. .
that no refrigerating machine can have a coefficient of performance
greater than that of a perfectly reversible machine working
the same two temperatures, and that all reversible refrigerating
machines working between the same temperatures have the same
coefficient of performance. Tt thus follows that the coefficient of
performance of any perfectly reversible refrigerating machine working
between temperatures T 1 and T 2 , whatever the nature of the working
substance, is equal to T 2 / (7*i  T 2 ), and this is the maximum possible.
A Carnot engine using perfect gas and working backwards will
be a most efficient refrigerating machine but its output will be
due to the small specil heat of gases. For large capacity we must
employ a liquid of large latent heat of vaporization and allow it to
evaporate at the lower temperature. We have seen in §6, p. 128
how practical considerations have led us to select ammonia and a
few other substances for this purpose.
34. Efficiency of a Vapour Compression Machine.— We shall
now calculate the efficiency of the vapour compression machine
HEAT ENGINES
[OJAP,
l\ ;
E A ^T
Fig. 23. — Cycle of a vapour
compression machine.
described on p> 127. Let us trace the cycle of changes which the
working substance undergoes. For
this purpose assume the valve V
to be replaced by an expansion
cylinder. AB (Fig, 23) represents
the evaporation of the liquid in the
refrigerator, BG the adiabatic
compression of the vapour by the
compressor, CD the liquefaction
of the sufjsLance inside the con
denser and DA the adiabatic ex
pansion of the liquid in the expan
sion chamber (not shown) . The
cycle ABCD is perfectly reversible
and hence its coefficient of perfor
mance is given by equation (20) .
In actual machines the throttle valve takes the place of the
expansion q f Iinder* This is done for mechanical simplicity. Conse
quently the part DA of the ideal cycle is replaced by DEA, On
account of die irreversible expansion through the valve there is a loss
of efficiency. The losses are twofold. An amount of work equal to
the area DEA is lost and there is also reduced refrigeration effect
since some liquid evaporates before reaching the refrigerator. The
coefficient of pei .: for the actual machine can be calculated as
the case of Rank' le (n, 220) from the values of the total
of the working sub; the different states.
Books Recommended.
i . \e, Engines.
fug, Steam Engine and other Heat En
Glazebrook. A Dictionary of Applied Physics, VoT. 1., Articles
on Steam Engine ; Engines, Thermodynamics of Internal Combus
tion : Turbines., Development of ; Turbines, Physics of.
4, Judge., Handbook of Aircraft Engines (1945) , Chapman &
Hall Ltd.
.1, Clarke, Interplanetary Flight (1952), Temple Press Ltd.,
London.
CHAPTER X
THERMODYNAMICS
1. Scope of Thermodynamics. — ■ The object of the science of
JTeinL is the stud) p£ all natural phenomena in which heat plays the
leading part. We have hitherto studied a few of these phenomena
in a detached way, viz./ properties of gases, change of state, varia
tion of heat, content, of a body (calorimetry) , There are other pheno
mena (e.g*., Radiarion, i.e., the production of light by heated bodies)
which have not yet been treated ; there are others which, strictly
speaking, do not come under physics, vis,. Chemical equilibria, Elec
trochemistry, but in which heat plays a very prominent part. The
detailed study of all these phenomena is beyond the scope of this
volume. The object of this chapter is simply to develo]
methods for the study of all such phenomena.
The first requisite for such studies is the definition of the thermal
state of a body or system of bodies. Next comes the development of
general principles* The studies fall under two heads* (a) the
study of energy relationships, (it) the study of the direction in which
changes take pi ace t The guiding" principles for the two cases are the
First and the Second Laws of Thermodynamics.
2. The Thermal State of a Body or System of Bodies.— The I
mal state of a simple hoi ius body like a gas or a solid is
defined by its temperature 7\ pressure p or the volume V, These
quantities are, therefore, called Tkermodynamical Variables. Of the
three quantities only two are independent, the third being automa
tically fixed by the value of the first two, for it is a common obser
vation that the pressure, volume or temperature of a substance in
any state, solid, liquid or gas, is perfectly definite when the other
two quantities are known. Hence for every substance the pressure,
volume and temperature are connected by a relation of the form
P = ffaT) (1)
This is called the 'Equation of State' of the substance. The
oE (1) is known only for a perfect gas (Chap. IV), But though (I)
is erally unknown we can proceed a good deal towards study
ins the behaviour of a substance under different conditions by using
differential expressions. In the subsequent work we shall occasionally
use differential calculus and some of its important, theorems. The
reader must clearly understand the physical interpretation of these
and for that purpose the following brief mathematical notes are
appended,
3. Mathematical Notes.— Suppose that y is some function of
the variable x and is continuous over a certain range of x. Let 8x be
some increment in the value of x and let Sy be the corresponding
238
THERMODYNAMICS
[CHAP.
Increment of y, such that y j &y still lies within the range of y con
tinuous* Then the value of the ratio £ as fix tends to zero is the
ox
differencial coefficient of y with respect to x and is written r .
ax
If x is a function of the two variables y and z., ue. t
* = / <y, *
there are two differential coefficients of the function. In the first
case the function may be differentiated with respect to 5? keeping z
constant, and in the second it may be differentiated with respect to
ping y constant. These differential coefficients are known as
parlial differential coefficients and are written
». I
where the suffix outside the bracket denotes the variable which is
kept constant during differentiation.
In order to find the variation in the quantity x when y and z
are both increased by the amounts Sy and &z we must make a double
application of Taylor's theorem. Now if x=f(y) we have from
or's theorem
(3*1 _
When x is a function of the two variables y and z, the value of
., z + &) may be obtained by first considering the second
varj it zf 8z and expanding in terms of Sy only
teorem. In the second stage y is kept constant and the
_ Ll  l) h are functions of zf Sz are expanded by Taylo >rem
in Ll  rii Let us denote the partial derivative of f(j, z) with
respect to v by /., and so on. Hence
=f[J\ .3:} H >, e+fc)
KW .,G?{.y 5 cfS^r) + higher terms in tj>.
Expanding each term on the righthand side by Taylor's theorem
in terms of S2 we obtain
+H<W/wCa *) +28>8c/, 4 o, «) + (&e)v«t>!» z)} + . (3)
If instead of expanding first in terms of Sv, we had expanded first
m terms of 82 and then in terms of 8y we would have obtained an
expression similar to (3) with the only difference that instead
t y t iy t z) Id now obtain jL , (y t z)> Since these two values of
ArfSx must be identical we have
MATHEMATICAL NOTES
or
(*)
i _ d~'.:
ie order in which the differentiation is performed is immab
If the variations 8y, gz are continuously decreased so that the
terms involving their higher powers may be neglected, equation (3)
reduces to the form
By/, +&£./(
or
&*
S*
is the theorem of partial differentiation.
Let us consider the equation
(5)
(6)
On comparison with (5) we see that if equation (6) is obtained from
an equation of the form (2) we should have
M
!."(&■
(7)
Conversely, iF M and A r are of this form there must exist a functional
relalion of the form (2) connecting the variables y and & When (2)
exists condition (4) h satisfied which from (7) is equivalent to the
condition
(8)
Bz
Hence if condition (8) exists, equation (6) must result from an
equation of the form (2) . Then Sx is said to be a perfect differential.
The physical meaning of this is that the change in x for the change
in the variables y and z from y L , z x to y 2 , z* is given by
*i  x 2 = f(y t , Zt)  /(V 2 . ZsO ,
and is consequently independent of the path of transformation,
depending only upon the initial and final values of the variables.
There exist some physical processes for which a relation of the
type (6) exists but (8) is not satisfied. Then §x is not a perfect
liferent ial. As a mathematical example consider the expression
2
Then
Sx « 3k dy ■ V 2fd&
dx „ Bx a
dj'dz dzdy
m
(10)
=
240
Tl I FRMODYNAMI CS
[CHAP.
e (9) cannot be integrated as can be easily verified, The condi
tion of integrability is just the reverse of (10) . If we multiply" (9)
by yz the resulting function 3y 2 z*dy j 2y ? 'zdz = dfy^z 2 ) is integrable,
The quantity yz is called the In fating factor of the expression
In equation (5), suppose z is kept constant, i,e. f §2 = 0. '1"
on division by $x, we obtain
or
1 UAUd,
which implies that the reciprocal of tile partial differential coefficient
is equal to the inverted coefficient
4. Some Physical Applications. — From equation (1) we have, on
differentiation, in the general case
*(&),■*+{&).":
(12)
For an i. so baric or isopiestic process, dp = 0. Hence equation (12)
yields
dp_
dT
) e U'/tU'
> /a
H
,r) —coefficient of volume expansion in an iso
o 1 I p
baric process (Chap. VII).
v 1^7) = bulk modulus of elasticity in an isothermal
process, or inverse of the coefficient of
if nx:?. nihility.
— l^l = coefficient, of pressure increase in. ar
choric (constant volume) process.
^Equation (13) shows that the volume elasticity, the volume co
efficient of expansion and the pressure coefficient of expansion are
not independent but are interrelated.
As an illustration, consider mercury at 0°C. and under atmos
pheric pressure. The coefficient of volume expansion is LSI >< JO 4
per °C. and the coefficient of compressibility" per atmosphere is
3.9 X 10* Hence from (13)
{ m
1 Tdv ]
v IdTl
\of !
v l dp \ T
cr = 4b 5 atm..
3»9xl0<
x.]
DIFFERENT FORMS OF ENliRGY
241
and the pressure coefficient
7fe*7."
465.
Hence it follows that a pressure of about 46 atm. is required to keep
the volume of mercury constant when its temperature is raised from
Q°C« to PC. Thus if the capillary tube of a mercury thermometer
is just filled with mercury at 30 °G., the pressure exerted in the
capillary when it is heated to 34 °G. will become 4 X 46\5 ^= 186
atmospheres.
5. Different Forms of Energy, — As thermodynamics is largely
a study of energyrelationships, we must clearly understand the term
'Energy'. The student must be familiar with the concept of 'Energy'
from his study of mechanics. 'Energy' of a body may be denned as
its capacity for doing work and is measured in ergs on the C.G.S.
system of units.
Everybody must be familiar with the two ordinary forms of energy,
viz., the kinetic energy and the potential energy. A revolving fly
wheel possesses a large amount of kinetic energy which can be
utilised in raising a piece of stone from the ground, Le,, in doing
work. Tn this process the kinetic energy of the flywheel is converted
into the potential energy of the mass of stone. The mass of stone
also in its raised level possesses energy which we call potential
energy and which can also be made to yield work. For when the
stone falls down under gravity the flywheel can be coupled to a
machine and be made Lo do work. Thus kinetic and potential ener
gies are mutually convertible and in a conservative system of forces
the sum of the kinetic and potential energies is constant throughout
the process. All moving bodies possess kinetic energy by virtue ol
their motion and all bodies placed at a distance from a centre of
force possess potential energy by virtue of their position.
There are other forms of energy also with which the student ol
physio must have become familiar, and which we shall now take
up. We have already pointed out that heat consists in the motion
of molecules and must therefore be a form of energy. The experi
ments of Joule and Rowland showed that there exists an exact pro
portionality between mechanical energy spent and the heat developed.
Friction is the chief method of converting mechanical energy into
heat. The conversion of heat into work has also been accomplished
by the help of steam engines and other heat engines which have
been completely described in Chap. TX. Thus it has been fully
established that heat is a form of energy.
^ Energy ma y be manifested in electrical phenomena also in a
variety of ways. It is well known that when two currentcarrying
coils are placed with their planes parallel to each other, they attract
or repel each other depending upon the direction of flow of the
current. In this case we may take the more familiar example of the
tramway. Here electric currents are made to run the carriage. Again
16
242
THERMODYNAMICS
[CHAP,
a wire carrying a current becomes heated Tints electricity is also a
form of energy which can be easily converted into mechanical or
thermal form."
Energy may also be manifested as chemical energy. For we know
that a bullet fired from a gun possesses enormous kinetic en<
which is derived from the chemical actions that take place when the
powder detonates. Further a large amount of heat is evolved during
a number of chemical reactions, viz., in the combination of hydrogen
and oxygen. Also the chemical aciions going on in a voltaic cell
produce electrical current. Thus chemical energy can be easily con
ed into mechanical energy, heat, electrical energy, etc.
There are various other forms of energy, all of which can be
converted into some one of the other forms. The important ones are
the following ;— magnetic energy, radiant energy (Chap XI) , surface
energy (surface tension), radioactive energy and energy of gravitation.
There are three important laws governing energy transforma
tion :— (I) The Transmutation of Energy, (2) The Conservation of
Energy, 0) The Degradation of Energy,
0. The Transmutation of Energy. — This principle states that
energv converted from one form to another, We have already
seen how mechanical energy, electrical energy and chemical energy
can be converted into heat. In fact all kinds of energy can be directly
.verted into heat. Mechanical energv obtained from a waterfall
irking a turbine, may be utilised to generate electrical energy by
Uphng the turbine to a dynamo. Electrical energv may be con
rted into thermal as in electrical furnaces, or into' chemical
?rgy as in t: secondary cells. Heat energy can be
ced into electrical energy as in the phenomena of thermoelectri
n to mechanical energy ; ces r and so on. Examples
couM be multiplied indefinitely.
The Conservation of Energy,— The principle states that during
• interchanges of energy, the total energv of the
tem remains constant. In other words energy is indestructible and
unbeatable by any process. No formal proof of the law can be given.
It is a generalisation based upon human experience and is amply
home out by its consequences. The law arose out of the attempts of
earlier philosophers to devise a machine which will do work without
expenditure of energy. They however railed in their attempt Later
it was recognized that al! the machines invented were simplv devices
to multiply force and could not multiply energy, and the true func
tion of n machine was to transmit work, Onlv that much work could
he obtained from a machine as was supplied to it, and that too in
idea cases. This shows that energy obeys the law of conservation.
In these cases, however, all energy cannot be recovered as some is
dissipated as heat But heat is also a form of energy, and on equating
the work obtained plus the heat developed to the\vork done, we find
that the relation of equality is satisfied. Thus the principle is estab
led for this case and follows from a result of human experience
which may be stated m this form:
x.]
FIRST LAW OF THERMODYNAMICS
243
"It is impossible to design a mitchine which will create energy
out of nothing and produce perpetual motion. Energy can only be
transformed from one form to another"
8. The Dissipation of Energy. — Energy, we have seen, is capable
of existing in different forms. In some forms however it is more
available to us than in other forms, Le t> we can get more work from
energy in one form than in another. For example, mechanical energy
is the highest available form while the heat energy of a body at a
low temperature is in a much less available form. ' Now the law of
degradation or dissipation of energy says that energy always tends to
pass from a more available to a less available form. This is called
the degradation or dissipation of energy, and is intimately connected
with the Second Law of Thermodynamics ; it will therefore be taken
up later.
FIRST LAW OF THERMODYNAMICS
9» The First Law* — We have alreadv enunciated the law of con
servation of energy. The First Law of Thermodynamics is simply a
particular case of this general principle when one of the forms of
energy is heat It may be mathematically stated in the form*
sc> = w f m,
(14)
where SQ is the heat absorbed by the system, S£/ the increase in its
internal energy and SW the amount of external work done by it. It
may be remarked that 3Q is here expressed in energy units, and can be
found by multiplying the heat added in calories by L the mechanical
equivalent of heat. For a simple gas expanding against an external
pressure p, SW = p8V, but for surfaces, magnetic bodies, etc, other
suitable terms are to be added to take into account other forms of
energy such as mechanical, electrical etc. The quantity 'internal
energy' is defined with the help of the principle of conservation of
energy by considering a thermally insulated system for which
8U = &W. Changes in internal energy can thus be measured in
terms of the external work done on the system.
Applications of the First Law
10. Specific Heat of a Body. — It is easy to see that the internal
energy of a body or a therm odynamical system depends entirely on
its thermal state and is Uniquely given, by the independent thermo*
dynamical coordinates. In the case of a simple homogeneous body
we have seen on p. 237 that, any two of the variables p, v f T are
sufficient to define its state uniquely. Choosing therefore V and T as
independent variables we have
U = f{V t T)> .
* It was first stated in this form by Gausius.
• OS)
244
THERMODYNAMICS
[CHAP.
where U and V refer to a grammolecule of the substance. Differen
tiating (15) we get*
f (#).«■+ (w) r * ■ • • ™
For a perfect gas LI —0 (p. 47); for a gas obeying van der
WaaJs* equation (see § 33)
IBU\ _ a
( 3 V I T F •
l£ an amount of heat 8.Q be added to a thermodynamical system*
say a perfect gas, part of it goes to increase the internal energy of
the gas by dV, while the remainder is spent in doing external work,
e.g., when the gas expands by a volume dV against the pressure p.
We have from the first law,
8Q=idUipdV t , (17)
where SQ, dU are expressed in energy units. Substituting for dU
from (16) in (17), we get
and dividing by dT we have
Hl_ldU\ r idu\ ]dV
SO
Now ■ j^= C, the grammolecular specific heat. If volume be
kept constant,
(#).(#).*■ •   w
from definition. If pressure be kept constant, I ~) =C b , the
V dT f p
specific heat at constant pressure. Hence
For a perfect gas { ?£) ^ 0 and p (^l) = E, hence using (18)
*It is thus seen that the quantity dV is a [perfect differential of a function
U at the variables dcfinmjf the slate of the bodv; we can therefore write dV
place of SU. _ Tn contrast with this the quantities, SQ, 8lV are not perfect
■ seen from the indicator diagrams since
differentials.
2 ."j 1 , 3 f" d^P*™ 1 "Pan the path of the transformation and not only upon
the initial and Jmal states. Hence we have denoted these changes by $Q and &W
Some people use 4 in place of these 3's hut then this d should "not" he taken
It J e d ?ff crcntial scnsc  *" is however a perfect differential and we can write
v\¥ = paV.
WORK DONE IN CKKTAIN PROCESSES
245
C p C; = R, ... * , (20)
as has been deduced on p. '16. The relation for solids and liquids is
complicated (see sec. 38, Example 1) .
It is easy to see that 8Q, the heal added, cannot be deter
mined only from the inilial and final states of the body, and a
ledge of how the heat, has been added is essential. Hence
C, the specific heat, has no significance unless the external conditions
are prescribed. We have defined C p and C , but there may be other
processes, e.g., adiabatic, when §(> = and C = 0.
11. Work done in Certain Processes. — The following expressions
for work done in different processes have been already proved : —
(a) Work done by a perfect gas in isothermal expansion is
(vide p. 215) equal to
RTlQg t (V & IV'.).
(b) Work done by a perfect gas in adiabatic expansion in which
the temperature falls from 7\ to T 2 (vide p. 215)
y— 1
(C)
12.
In the case of a cyclic process the substance returns to its
initial state and hence dU = 0. Equation (17) then yields
dQ_ = pdV/j, i.e., the heat absorbed by the substance is
equal to die external work done by it' during the cyclic
process. Thus in Carnot's cycle the work done during a
cycle is Ql  (£.,
Discontinuous Changes in Energy — Latent Heat.— When a
body is in the condensed state (solid or liquid) and is subjected to
increasing temperature, the state may suddenly change from solid to
liquid form (fusion), or from liquid to gaseous form (evaporation) , or
from one crystalline form to another (allotropic modification) . In
such discontinuous changes the energy also changes discontinuously
(phenomena of latent heat). Let us consider the evaporation of 1
gram of water at 100°G, and 1 atuL pressure. The application of
First Law yields
L = u 2 u 1 \p{v z v 1 ),
(21)
where %, v 2 respectively denote the internal .energy and volume of
1 gram of vapour, and %, Vj_ the corresponding quantities for the
liquid.
Equation (21) expresses the fact that the heat added is spent in
two stages : (1) in converting 1 gram of water to 1 gram of vapour
having tile same volume, (2) in doing external work, whereby the
vapour produced expands under constant pressure to its specific
volume.
246
ISKRMODYNAMICS
; CHAP.
u^ — Ui is sometimes known as the internal latent heat and
p(v s — Vj) as the external latent heat. For water at 100°C,
Vi = 1 tc, v 2 = 1674 c.c. (specific volume of saturated vapour) ,
„ .. . 1013 xlO»x 1673 . ..„ ,
Hence p(v t  v{\ — n= — ^xi cal. sz 40.5 cal.
1 N ' 4lBx 10 7
Since L = 538.7, the Internal Latent Heat « 2  «i = 498.2 Cal,
It is often convenient to denote the different states of aggregation
of the same substance according to the following convention, [H 3 0],
(H 2 0), H a O denote water in the solid, liquid and gaseous states
respectively. The quantity taken is always a mol. unless otherwise
stated. The symbolical equation
H a O = (H a O) J 10,170 cal ... (22)
expresses the experimentally observed fact that the energy content,
of a mol. of H 2 (gas) at 0°G exceeds the energycontent of a mol
o£ H.O (liquid) at'0°C. by 10,170 calories, die change taking place
at constant volume.
We have further
(H 2 0) = [HaO] + 1^30 cal, . . (23)
us, t energy of a mol. of liquid II s O at 0°C. exceeds the energy of a
moL of ice at 0°C. by 1,430 cal
From equations (22) and (23) we have by addition
H s O = [H..O] + 11,600 cal.
i.e., heat of sublimation of 1 mol. of ice at constant volume is 11,600
calories. This is a consequence of the First Law applied to a physical
process and can be easily verified,
13. The First Law applied to Chemical Reaction* — Hess's Law of
Constant HeatsommatioiL— The First Law can be extended to Chemi
cal Reactions, Thus when 2 mols, of H 2 and 1 moL of 2 are
exploded together in a bomb calorimeter, 2 mols. of (H a O) are formed
and 136,800 calories of heat are tvolved. Allowing [or the heat
evolved in passing from the gaseous to the liquid state, we obtain
from the First Law, since SQ =  116,500 and BW =0,
2t/ HQ + Po.aOW.o = H6,500 cab
!>.., the energy of 2 mols. of H. and 1 mol. of Or, exceeds the energy
of 2 mols, of 'H a O by 116,500 calories,
Hess stated in 1840, before the First Law had been discov i :
that if a reaction proceeds directly from state 1 Co state 2, and again
through a seri: o£ intermediate states then the heat evolved in the
direct change is equal to the algebraic sum of the heats of reaction
in the intermediate stages. This is an important law in Thermo,
chemistry and is known as Hess's Law of Constant Heat Summation.
The law is illustrated by the following examples : —
G (diamond) + O a = C0 2 f 94,400 cal
CO+ £O s = CO a + 68,300 caL
x.
SECOND LAW OF THERMODYNAMICS
241
We have by adding
C (diamond) + 2 = CO s J 94,400 cal.
This is verified by experiments.
We take another example. Let the symbol (LiOH) m denote the
solution : ol. of the substance. Then take the equations:—
2 [Li] 4 2 (H 2 0) = 2 (LiOH) an f H s + 106,400
2[LiH] f 2 (H s O) = 2 (LiOH)*, f 2H 2 + 63,200
The energy evolved is very easily determined from solution ex
periments. Subtracting the lower equation from the upper one •
obtain
2[Lil f H* = 2[LiH] +43/200 cal.
i.e., the heat of formation of two mols of [LiH] from [Li] and Tl 2
is 43,200 cal,
THE SECOND LAW OF THERMODYNAMICS
AND ENTROPY
14. Scope of the Second Law.— The Second Law of Thermodyna
mics deals with a question which is not at all covered by the First
Law, viz., the question of the direction in which an}' physical
chemical process involving energy changes takes place.
A few illustrations will clear the point. Let us consider tru
thermodynamical process illustrated by the symbolic equation :—
2H 2 f 2 = 2H 2 + U calories.
The equation tells us that if 2 mols. of H 2 gas combine with o
mol of O s , 2 mols of H.O vapour are formed and U calories of heat
are evolved, or vice versa, when 2 mols of H 2 Qvapour are decompa
completely into H ; , and 2 , U calories of heat must, be a', irh
This result is obtained as a matter of experience from ealorimetric
experiments, and we interpret the result according to the First Law
of Thermodynamics by saying that in the combination of 2 molecu
of hydrogen with 1 molecule" of oxygen to form 2 molecules of H s O,
[Tie diminution, in energy amounts to JU/N ergs. This can
be further utilised in calculating energy relations in other reactions.
But the first law cannot tell us in which direction the reaction will
take place. If we have a mixture consisting of H 2J O a and H a Ovaponr
in arbitrary proportions at some definite temperature and pressure,
i lie first law cannot tell us whether some H 2 and Q 2 will combine to
form H ; .0 or some H 2 will dissociate into H 2 and Q 2 , the system
i hereby passing to a state of greater stability.
Or we can take a physical example. When two bodies, A and B.
exchange heat, the first law tells us that the heat Iosl by one body
is equal to the heat gained by the other. But it does not tell us in
which direction this heat will flow. Only experience tells us that
heat will pass from the hotter body to the colder one spontaneously.
248
THlUtMODVNAMlCS
[CHAP.
but in speaking of hotter and colder bodies we are making use of a
V physical concept (temperature) which is not at all included with
in die scope of the first law. Tor giving us guidance in deciding the
question of direction in which a process will take place, we require
a new principle and this principle, which arose out of Carnot's specu
lations about the convertibility of heat to work, is genially known
as the Second Law of Thermodynamics.
15. Preliminary Statement of the Second Law. — The problem of
convertibility of. heat to work has been treated in Chap, IX. It is
readily observed that the main question there was about the direction
of energy transformation. Mechanical energy and heat are only
different forms of energy, but while mechanical energy can be com
pletely converted to heat by such processes as friction, it is not possible
to convert heat completely to work. Even by using reversible engines
which are the most efficient, only a fraction can be converted to
work. The conversion is therefore only partial. The question is,
why is it so ?
An answer to this question has been given already in Chap. IX,
viz., if it were possible to design an engine more efficient than a
reversible engine, we could continuously "convert heat to work, and
this will produce perpetual motion of a kind. We may call (his
perpetual motion of the second kind. We are convinced that this is
not possible, and we may start from a statement which expresses
our con vie Lion just as the first law is based on the conviction that
Energy cannot be created, but can only be transformed. We may
ateirfent In the following form: —
. impossible to construct a Heat Engine which will con
ly abstract heat from a single body, mid 'convert the whole of
:he working system," This
ement is equivalent to Clausius or Lord Kelvin's" statement of die
Second Law.
Lord Kelvin stated the law in this, form : — '7* & impossible by
/ inanimate material agency to derive mechanical effect from
■ lion of matter by cooling it below the temperature of the
lest of the surrounding objects,"
Clausius staled it in the form : — "It is impossible for a selfacting
machine, unaided by any external agency, to convey heat from one
body to another at a higher temperature,' or heat cannot of itself pass
from a colder to a warmer body."
It must be clearly understood that these statements refer To con
tinuous or cyclical processes. For it is always possible to ]et heat
pass from a colder to a hotter body and at the same time the sub
stance may change to a state different from its in i tin! one. As an
example consider an isolated system comprised of a cold compressed
gas and a hot rarefied gas communicating with each other through a
movable piston. When the piston is released, the hot vapour is com
pressed and heat is thereby transferred from the cold to the hot gas.
The state of the two gases however changes as a result of the process.
To face p. Z4S
Lord Kelvin (p. 248)
William Thomson, Lord Kelvin of Largs, born on June 26, 1824, in Belfast
died qn December 17, 1907, His important contributions to Heat
arc the Second Law, the absolute thermometric scale and
the JouleKelvin effect. He carried on valuable
researches in Electricitv.
Rudolf Cuaustus (1822 (p. 248)
at KosKn, he studied in Berlin and became Professor of Physics
successively at Zurich, Wureburg and Bonn. Simultaneously with
Lord Kelvin he announced tJie Second Law of Thermo
dynamics. He was one of the founders of the Kinetic
Theory of Gases.
X.]
ABSOLUTE SCALE OF TEMPERATURE
219
16. Absolute Scale of Temperature.— Lord Kelvin showed in 1881
that with the aid of the ideal Camot engine it was possible to define
temperature in terms of energy, and the scale so obtained is inde
pendent of the nature of any particular substance, £ Kelvin started
the ret.uk (p. 219) that' the efficiency of all reversible engines
ing between two temperatures is a function of the two tem
peratures only and is independent of the nature of the working sub
stance. This may be mathematically stated in the form
W
Q*
/(*!> ^
(24)
where W is the work done by the Camot engine, Qt the heat taken in
and i3 6 2 tile temperatures* between which the engine works.
Now W= QiC&f where Q* is the heat rejected. Hence
^
=/&, *■).
or
ft.
 F(e lt e t ),
where F denotes some other function of L and Q%.„
It can easily shown that the ratio Qi/Qa can be expressed
in the form «/f(0i)/^ (&i) ■ "For if we have reversible engines working
between the pairs of temperatures (0 a , d b ), {$$, 3 ), (B u 3 ) and the
heat absorbed or evolved (as the ease may be) at these temperatures
be <b> Gap gs, then
^— m, 0*},
Q
n.
^ = FiB,, 6,), £  F(0 1; 3 ). . (25)
ft. ^ *" ft* <
From these by multiplying the first two expressions we get
(L
= F(B lt fl B ) X F(B tl 3 ),
Hence the function F must satisfy the relation
F\9 X , 9 a ) = F(9 U 9 t ) X F%,
This will be so only when F(Q it d £ ) is of the form ^(0, )/${#,
fore for any reversible engine
ft* *&) " '
It will be seen that i & 1 >tf 2( C>]>f.> 2 and therefore 'P(0 1 )>'ii(6 i ) )
i.e., i/>(0) is a quantity which increases monotonically with Q and may
be used to measure temperature. Denote the value or ip (0) by t.
The relation (27) then becomes
0,1 *i
(26)
There
(27)
&
(28)
* These may be measured on any arbitrary scale.
250
TJIERMODYNAM ICS
CHAP,
Equation (28) is used to define a new scale o£ temperature t which
is tailed the absolute or the thermodynamic scale. This scale does
not depend upon the properties of any particular substance for equa
tion (28) is universally true. The ratio of any two temperatures on
this scale is equal to the ratio of the heat, taken in and neat rejected
by an engine working reversibly between the two temperature. 1 ;.
The zero of this scale (r=0) is that temperature at which
(from 28) and hence Ti r — (^, Thus all the heat taken by
the engine has been converted into'work and the efficiency of the
engine is unity, t cannot be less than this, i.e., negative, for if it
were so, Q 2 would be negative which implies that the engine would
be drawing heat both from the source and the sink. This is impos
sible from the second law and hence t = is the lowest temperature
conceivable. This is the absolute zero of temperature which need
not be the temperature at which all molecular motions cease to exist
as one might otherwise infer from elementary kinetic theory.
The zero having been determined, let us fix the size of the degree..
In conformity with general usage we suppose that the interval be
tween the freezing and the boiling points of water is divided into 100
equal parts on this scale, viz, f
3*.
hioo
(29)
The thermodynamic scale has thus been completely defined and fixed.
scale can be actually realised in practice.
For this we must investigate the property of some actual sub
stance. A perfect gas is very suitable for this' purpose. In fact we
have already shown on p. 216 that for a Carnot engine using a perfect
gas as the working substance,
. (SO)
where T lf T z are the temperatures measured on the perfect gas scale.
Equation (28) combined with (30) yields
(81)
*",
'.:.. the ratios of any two temperatures on the thermodynamic
and the perfect gas scale aTe equal. Thus the absolute
narnic scale coincides with Che zero oC the per!) , ■ ■. ther
mometer; for if t o — O, r 2 = 0. Further from (29) and (31) since
the Interval between the ice point and steam point is 100° on both
scales, r, e , and T iM are identical, and in general from (31) other
temperatures on the two scales are also identical, Thus the thermo
dynamic scale is given by the perfect gas thermometer,. For the
methods of reducing the ordinary gas scale to perfect gas scale
p. &
x.)
ENTROPY
ENTROPY
251
17, Definition of Entropy.— The formulation of the Second Law,
as on p. 248, is not of much use for finding out the direction of
change in a chemical or physical process. We want a more general
and mathematical formulation which was supplied by Clausius.
Clausius shifted the interest from the problem of convertibility
of heat to work, to that of change of thermal state of the working
substance, for it is only the working substance which undergoes a
thermodynamical change in the process. He observed that when the
heat Q is added to it reversibly at temperature 7\ and £>' taken away
from it reversibly at temperature T r we have
a
T
a; a
(32)
This property was utilised in defining an additional thermodyna
mical function (Entropy) for the gas which proved to be of great
importance in the further development of thermodynamics. In the
following deduction, we have followed Swing's treatment.*
Let us take a gas in a definite state A (Fig. 1), and compare this
state with another state B, which is obtained by the addition of heat
energy, Jt is easy to see that the
amount of heat added from outside may
be anything, owing to the variety of
ways in which heat can be added. This
can be visualised by drawing curves be
twecn A and B. There may be an in
finite number of curves passing between
A and B, corresponding to the infinite
number of ways in which heat can be
added and for all these the amount &Q
of heat added will be different. It will
be shown, however, that the integral
measured along any reversible process does not depend on the path,
but only on the coordinates of A and B.
Let us draw two adi aha tics through A and B and draw an isother
mal DC at temperature t cutting these adiabatics in D and C respect
ively. Let us now divide the area A BCD into a number of elemeutai'V
Carnot eyries #yyV as in Fig. 1 (xy, x'y r , isothermais, and xx', yf
adiabatics).
Let SQ be the amount of heat absorbed at temperature T while
*Sce Glazebrook, A Dictionary of Applied Physics, Vol. 1, p. 271.
 vC
V
Fig. 1. — The entropy
function.
252
THERMODYNAMICS
[CHAP.
*]
ENTROPY OF A SYSTEM
253
the point shifts from x to y'and Sq be the amount given up while it
shifts from / to x'. Then we have
HI
7 t '
Adding up for all the Carnot cycles between A and B, we have
M
r 4T*
m
This is the value of
along the zigzag path Aabxy, B, for
the value of the integral along the adiabatics Aa, bx, ....is 0.
In the limiting case when the Carnot cycles are infinitely small
the zigzag path Aabx B coincides with AB,
Hence
I
Iff=><
(34)
r B so
The value of die integral j 2? is now seen to be equal to
'XBq/t and hence it is independent of the path between A and B.
Instead of taking the particular isothermal DC we may draw any
other isothermal cutting AD and BC, the value of the integral cannot
be thereby changed, for if we construct a number of Carnot cycles
between the first and the second isothermals we have
1 ' * I ... .
 S8 ? = , E&q*.
The points A and 1 ! taken anywhere on the adiabatics AD.
and BC respectively, the value of the integral will remain the same,
for J ^ along an adiabadc is zero since S£> = Q. Hence it follows
that if we pass from one adiabatic to another the expression I — =*
J % T
denotes a quantity which increases by a fixed amount independent of
the manner of transformation. If we start from some zero state and
denote the value of the integral f £ by s A it follows that
Jo ■*
function of the coordinates of A alone. Clausius called it the entropy
of the substance. Now since we are not familiar with the zero state
of the body when it will be deprived of all its heat, we cannot, find
in this way the absolute value of the entropy in any state. We arc
however generally concerned with changes in entropy and hence we
can measure from any arbitrary zero as is done in the case of energy.
To measure the entropy m any state we have to take the substance
along any reversible process to its zero state and find the value of
lAq/t along this process. The entropy, though conveniently meaaur
jj. is a
ed with respect to a reversible process, has nothing to do with it and
exists quite independently of it.
The above treatment is perfectly general and w r ill hold whatever
substance is taken as the working substance. Now if the work is
done by expanding against an external pressure we have from the first
law for any process reversible or irreversible, &Q = du \ pdv, where p
stands for the external pressure at every stage o[ the process. But
for finding entropy we have to substitute the value of Sf> along a
reversible process. Now the process is reversible only when "the
external pressure p is always equal to the intrinsic pressure of the
substance, i.e., when the expansion is balanced. Hence we get*
J = j r J«+M' m
where p denotes the pressure of the substance itself.
It is easily seen that entropy is proportional to the mass of the
substance taken, for if we take M grams of the substance dU — Mdu t
dV =z Mdv and hence 5 Ms.
Exercise. — Find the increase of entropy when 10 grams of ice
at 0°C. melt and produce water at the same temperature, given that
the latent heat of fusion of ice = SO calories/gm.
Since the process is reversible and isothermal we have
AS = i£ 3 — =2.93 caL/degree C.
18. Entropy of a System* — In the last section it was shown that
if the entropy per unit mass of a substance is s the entropy of m
grams of the substance is ms. Tt can be shown in a similar way that
if we have a system of bodies in thermodynamical equilibrium with
different thermodynamical variables and having masses m 1t nu. . and
specific entropies s s , s 2 . . ., the entropy of the whole system "is
S = »»!*! + m 2 s 2 + ... . . . . (30)
19, Entropy remains Constant in Reversible Processes.— The sum of
the entropies of all systems taking part in a reversible process remains
constant. Considering the Carnot cycle we notice that the working
substance has the same entropy at the end of the cycle as at the
beginning since it returns to the same state. The loss in em
of the source is Qi/Ti, the gain by the condenser is Qj/.T* and
hence the net gain in the entropy of the system is
T B T L  U < 3 '>
Thus the entropy of the whole system remains constant.
* A more general definition is the following : — If from any cause whatever
the unavailable energy of a system with reference to another system at TV under
goes an increase A£, then AE/7V measures the increase of entropy of the
system.
254
1H KRMODYN AM I CS
[chap.
In the isothermal expansion of the working substance in the first
part of the Carnot cycle the increase in entropy of the working sub
stance is (li/T u the loss by the source is Ql/T\ and hence the total
entropy remains constant. This holds for any reversible transforma
tion*
Working out as on p. 215 we see that the increase in entropy
of a grammolecule of perfect gas in isothermal expansion from volume
F 3 to V 2 at temperature T is
but by
The change in entropy during a reversible adiabatic compression
or expansion is zero. For in such a process the external work
done is I pdV and change in entropy is /\S = \ ^ —
the condition of the process from the first law dU\pdV =0, hence
the entropy remains constant in an adiabatic process. Adiabatic
curves are therefore sometimes known as "Iscntropics."
20. Entropy Increases in Irreversible Processes. — The entropy of a
system increases in all irreversible processes. As examples of such
processes we may mention the conduction or radiation of heat, the
rushing of a gas into a vacuum, the interdiffusion of two gases, the
opment of heat by friction, flow of electricity in conductors, etc
<vt the theorem here for a few cases.
Entropy during Conduction or Radiation of Heat,
— Suppose a small quantity of heat (? is conducted away from a body
A at temperature T, ro another body B at temperature T^{TiyT 2 ) ,
and I'. so small that the temperatures Tj and T 2 of the bodies
arc not appreciably altered in the process, then A loses entropy
equal to , r ~ while B gains entropy by an amount J?
gain in entropy of the two bodies is
Thus the total
**=**(* it)
m
which is positive since Ti>T 2 . Hence the entropy of the system as a
whole increases during thermal conduction. Similarly in the case of
radiation, if a quantity of heat SQ is radiated from the body A to the
body B, the gain in entropy is given by the same expression. Thus
increase of entropy is always produced by the equalisation of tem
perature.
(ii) Gas rustling into a vacuum. — Consider a perfect gas in a
vessel rushing into an evacuated vessel and let the whole system
be isolated. Since the gas is perfect the temperature does not change
* In denotes natural logarithm.
X.
ENTROPY OF PERFECT GAS
255
in the process. The gain in entropy may be obtained by finding the
value of I =■ along any reversible path connecting the initial and
final states. The reversible process most convenient for this purpose
is an isothermal expansion of the gas against a pressure which is
.always just less than the pressure of the" gas.
For a perfect gas dV at constant temperature is zero,
gain in entropy
A$=4rf pdV = Rln A
Hence the
(40)
where P it F 2 denote the initial and final volumes of the gas. Hence
the entropy of the system increases in this irreversible process.
21. Tne Entropy of a Perfect Gas. — Let us take m grams of a
perfect gas having die temperature T and occupying the volume v.
The entropy ms is given by the relation
7" durpdv
o r~~ *
Since du = mc„dT, p=zmRT/Mv f we have
ms —
ms = m (c v In T 4 rj In v) 4 const.
. (41)
We can also express wis in terms of pressure. From the relation
R
M
tp  e w r
wet get ms = m(c p In T — ^ In p) \ const.
(42)
Exercise. — Calculate the increase in entropy of two grams of
oxygen when its temperature is raised from <J D to 100°C and its
volume is also doubled,.
&S =2x23026
5035
log
373
273
Second
+
I
log 2 =0184
cah
32 "*" 273 ' 32 ° j deg
22. General Statement of Second Law of Thermodynamics, — 'We
have shown above for a few irreversible processes that increase of
entropy takes place during such a process. This result is however
very general and holds for any process occurring of itself. We shall
not, attempt to prove here this general statement, but only enunciate
it in the following words : — Every physical or chemical process in
nature takes place in such a way that the sum of entropies of all bodies
taking part in the process increases. In the limiting case of a rever
sible process the sum of entropies remains constant. This the most
general statement of the Second Law of Thermodynamics and is iden
256
THKRMGDYNAMICS
[CHAP.
*•]
PHYSICAL OONCKPT OF ENTROPY
257
tical with die Principle of Increase of Entr* viz,, the entropy
a system of bodies tends to increase in all processes occurring in
nature> if we include in the system all bodies affected by the change.
Clausius summed up the First Law by saying that the energy
of the world remains constant and the Second Law by saying that
the entropy of the world tends to a maximum* Though the terras
'energy of the world' and 'entropy of the world' are rather vague, still,
if properly interpreted, these two statements sum up the two laws
remarkably well.
23. Supposed Violation of the Second Law.— Maxwell invented an
ingenious contrivance which violates the Second Law as enunciated
above. Following Boltzmann's idea, he imagined an extraordinary
being who could discriminate between the individual molecules. Sup
pose such a creature, usually known as Maxwell's demon, stands at
the gate in a partition separating two volumes of the same gas at the
same temperature and pressure and, by opening and shutting die gate
at the proper moment, allows only the faster moving molecules to
enter one enclosure and the slower molecules to enter the other
enclosure. The result will be that the gas in one enclosure will be at
a higher temperature than in the other and the entropy of the whole
system thereby decreases, though no work has been done. This
apparently violates the Second Law,
We observe, however, that to Maxwell's demon the gas does not
appear as a homogeneous mass but as a system compound o[ discrete
molecules. The entropy does not hold for individual mole
cules, but. is a statistical law and has no meaning unless we deal with
matter in bulk,
24, Entropy and Unavailable Energy.— Consider a source at
temperature Ty and suppose it yields a quantity of heat (> to a work
ing substance. If the lowest available temperature for the condenser
of the Carnot cycle is T n the amount of heat rejected by the Carnot
engine working 'between T t and T is QT /T V The remainder, i.e,
Q— Q=? has been converted into work. Thus the available energy
i s Q\ * — J ) ■ Now suppose a quantity of heat $() passes by conduction
from a body at. temperature T 2 to another at temperature TV The un
available energy initially was (T iY /T.,)Hl and finally it was (T /r 3 )SQ.
Hence the gain in unavailable energy is SQ \Y"" t) T °' ^° bemg
the lowest available temperature. The increase in entropy of the
system is, as we have already proved, equal to SQ Isr y). Hence
the increase of unavailable energy is equal to the increase of entropy
multiplied by the lowest temperature available. Thus entropy is a
measure of the unavailability of energy, and the law of increase of
entropy implies that the available energy in the world tends to zero.,
Le<, energy lends to pass from a more available form to a less available
form. This is the law of degradation or dissipation of energy mention
ed on p. 243 and is thus seen to be equivalent to the Second Law* It
follows as a corollary that all transformations, physical or chemical,
involving changes of energy will cease when all the energy of the
world is run down to its lowest form.
25. Physical Concept of Entropy,— The entropy of a substance
is a real physical quantity which remains constant when the sub
stance undergoes a reversible adiabatic compression or expansion. It
is a definite singlevalued function of the Lhermodynaraical coordinates
defining the state of the body, viz., the temperature, pressure, volume
or internal energy as explained on p. 237. It is difficult to form a
tangible conception of entropy because there is nothing physical to
represent it ; it cannot be felt like temperature or pressure. It is a
J statistical property of the system and is intimately connected with
' the probability of that state. Growth of entropy implies a transition
from a more to a less available energy, from a less probable to a more
probable state, from an ordered to a less ordered state of affairs. The
idea of entropy is necessitated by the existence of irreversible pro
cesses. It is measured in calories per degree or ergs per degree.
26. EntropyTemperature Diagrams. — We have represented the
Carnot cycle and other cycles in Chap. IX by means of the usual in
dicator diagram in which the volume denotes the abscissa and the
pressure denotes the ordinate. Another way to represent the cycle,
which is often found very useful, is by plotting the temperature of the
working substance as ordinate and the enixopy as abscissa. Its im
portance is readily perceived for during any reversible expansion the
increase dS in entropy is given by
dS = <f,
where the integration is performed
between the limits of the expansion.
Thus the area of the curve on the
: 1 1 iropyleniperature diagram* represents
the amount of heat taken up by the
substance.
The Carnot cycle can now be easily
represented on the entropytemperature
diagram by Fig. 2. AR represents the
isotlicrmal expansion at T\, BG the
reversible adiabatic expansion, CD the isothermal compression at
T 2 , the temperature of the sink, and DA the final adiabatic
compression. It is evident that lines AB and CD will be
* This is also called iephiyram as the symbol <f> was previously used for
entropy,
17
D
T*
5!
I*
* Entrtfpy b
Fsg._ 2,— Entropytemperature
diagram of Carnot cycle.
258
THERMODYNAMICS
[CHAP.
parallel to the entropyaxis and BC, AD will be parallel to the tem
peratureaxis. The amount of heaL taken in is represented by the
area ABba, heat rejected by the area DCbu, the difference ABCD
being converted into work. These areas are respectively A (.$'  S),
T 2 (&  5) , (T t  T s ) (S f  S) and the efficiency is 1  r 2 /7Y
27. Entropy of Steam. — For enabling 1 the student to have a pro
per grasp of the conception of entropy we shall calculate the entropy
o£ steam, Let us start with 1 gram of water at, 0°C. and go on adcf.
trig heat to it. The increase in entropy when we add a small amount
Of heat 8Q at T 6 is p , But. SQ == vdT where o is the specific heat
■of water at constant pressure. Hence the entropy of water at T° is
, cT ff dT . T
if tr is assumed to be constant. s denotes the entropy of water at
9°C) {T (1 D K.) and it is customary to put it equal to" zero. Hence
for water
 o log, ,
This is represented bv llie
where o can he put equal to unity.
logarithmic curve OA in Fig, 3.
If the liquid is further heated it boils at the temperature T x ab
its latent heat L. Since
the temperature remains constant
in this process the increase in entropy
per gram is L/T v This is repre
sented by AB in the figure. Hence
the entropy of 1 gram of dry satu
rated steam is
If
the steam is wet, q being its
dryness, i.e., q is the proportion of
dry steam in the mixture expressed as
a fraction of the whole, the entropy
of wet steam is
Tf the steam is superheated to
T.. c the entropy is further increased
by the amount
P
dT
where
Entropy
Fig. 3. — Entropytemperature
diagram o£ steam.
denotes the specific heat of steam
during superheating. This is repre
sented by BC in the figure. Hence
x.l
APPLICATION OF THE TWO LAWS
259
the entropy oL superheated steam at 7V
= <? 1°S*
n
i + r
IS
*P
dT
T
As c* v
273 ' T x j Tl
,aries with temperature, its average value over small intervals
this way the entropy of steam in any state can be
is talten. In
calculated.
Exercise. — Find the entropy of saturated steam at a pressure of
74 lb. per sq. in.
From the table the boiling point at this pressure is 152.G°C. and
the latent heat at this temperature is 503.6 cal./gm. The entropy
of 1 gmt of water at 152,fi°G or 425.6°K is 1 X log, 425.6/273 = 0.44
c&L/degree. The increase in entropy due to the evaporation at
425!g°K is 503.6/4256^ 118 cal./degree. Therefore the entropy of
steam at 425.6°K is given bv .44 f 1,18 = 1.62 cal./degree. This may
be verified by looking at the steam tables. For more accurate calcu
lation the variation of the specific heat of water with temperature
should be taken into account.
APPLICATIONS OF THE TWO LAWS OF
THERMODYNAMICS
28. General Consideration*.— For making frequent and effective
use of the Second Law, it should be expressed in a convenient form.
For this purpose a mathematical formulation is more desirable and
was supplied by Clausius (p, 252) , was.,
&Qr<fe* ( 4 *)
where dQ is the heat absorbed during a reversible process connecting
the two states. From the first law
$Q=du + pdv, (7)
where p denotes the intrinsic pressure of the substance and is given
by its equation of state. Combining these two equations we get
du + pdv=Tds (44)
Let us further investigate the nature of the variations du, dv
and <fo. As already stated in Chap. TV and also on p. 237, tire state
of every substance is fixed if we know any two of the variables p, v,
T, U....L For example, the internal energy uz=f(v, T) is uniquely
defined it v and T are given. Thus if we represent the internal
energy u of the substance" on the coordinate svstem v, T , the value
of u for every point is fixed and does not depend upon the path along
which we bring the substance to that point (state) ; in other words,
the total change du has the same value whether we first make the
change dv in v'and then the change dT in T or in the reverse order.
* The results deduced from this equation in the following pages i and their
confirmation by experiments may be taken as direct verification of this equation
and therefore: of the Second ! ,r>\".
260
THERMODYNAMICS
CHAl*.
Mathematically, this means that du h a perfect differential and as
explained on p. 239 we must have
O'u
a a — V7S (45)
BxBy djdx
where x and y are the two variables defining the state of the system.
The entropy function was shown on p. 252 to be such that its
value does not depend upon the path of transformation, i.e, f s is also
a singlevalued Junction of the coordinates. Similar is the case with
the volume and the internal energy.
As a contrast and for clearer understanding, it may be mentioned
that pdv and lQ= du \ pdo are not perfect differentials for their
values depend upon the path of transformation. This may be easily
seen as T pdv represents th^ area of the indicator diagram and has
different values for different paths. But though du \ pdv is not in le
viable we have proved by physical arguments that (du f pdv) /T is
in tegrable ; in other words, 1/7* is the integrating factor to the
energy equation.
We now proceed to make use of the above ideas for the further
development of equation (44) .
29. The Thermodynamkal Relationships (Maxwell).— Equation (44)
may be written as
du = Tdspdv. .... (46)
Now if x and y are any two independent variables we have, since «,
s and v are expressible in terms of x and %
, Bs Bs ,
j dv , , do ,
dv — ■=— ax ■ ay,
Bx By ■"
and
du =
du , . Bit ,
Substituting the values of ds, dv, du in (46) and equating the co
efficients of dx t dy, we get
<?«_ ~Bs _ dv
Bx~ Bx ~ ** dx'
Bv_ T ds _ dv
"By 'dy p dy
Now, since du is a perfect differential, we should have
d*u _ _ B'ht
BxBy~dyBx'
B ( r ds jv\ b 1 . r ds dv\
•
X.
FIRST RELATION
BxBj
Expanding
dydx
out and remembering that
we get
B*i> B z v
dxdy dydx
261
and
(4?:
er ds _bt B± = &p<h_ bp b^
dx" By By "Bx 3x By By Bx'
which when geometrically interpreted means that the corresponding
elements of area whether represented on T, s or p, v coordinates are
equal. This relation will hold for a simple homogeneous substance.
For convenience of remembering we may write (47) as
B{T*s) _B\p : b)
d(x,y) 3{x,yj '
where ' 'stands for the determinant
B{x t y)
BT
BT
Bx
dy
Bs
ds
Bx
By
Any two of the four quantities T, s, p, v can be chosen as x t y.
This can be done in six different ways and correspondingly, we have
six thermodynainical relationships, though all of them are not
independent.
30. First Relation, — Let us take the temperature and the
volume as independent variables, and put x = T, y = v in (47).
Then
BT _ , Bv_ _ .
Bx By
and
$1
By
dv
P = o,
Bx
(48)
since T and v are independent. Hence we have
(*) ={%) ■ ■ ■ ■
\ to/ r \3r } v
which implies that the increase of entropy per unit increase of volume
at constant temperature is equal to the increase of pressure per unit
increase of temperature when the volume is kept constant. We can
apply equation (48) to the equilibrium between the two states of the
same substance. Multiplying both sides by T
to It \dTf v
(49)
which means that latent heat of isothermal expansion is equal to the
product of the absolute temperature and the rate of increase of
pressure with temperature at constant volume. Thus, if a body
262
THERMODYNAMICS
CHA1\
changes its state at T Q and absorbs the latent heat L, and the specific
volume in the first and the second states are v x , v 2 , equation (49)
yields
L _ dp_
v*~v. at
or
dp_
dT
(50)
This is Clapeyron's equation and is one of the most useful
formulae in thermodynamics, A more rigorous proo£ of this will be
given in sec. 39.
Let. us first employ (50) to find the change in the freezing point
of a substance by pressure. In the case of water at o C 1
£, = 79.6 X 4,18 X 1A T ergs per gm.
T = 273.
1*2 = 1.000 e.e. (specific volume of w r ater at 0°C),
0!= 1.091 c.c. ( „ „ „ ke at 0°<
dp__ 796 x4l 8x10
'* dT 27S x (11091)'
Now if dp — 1 atmosphere = LOIS X 10 6 dyne/an 2 ,
8T= 0.0075°.
This shows that the melting point of ice is lowered by increase
of pressure, the lowering per atmosphere being 0.0075°. The pressure
isary to lower die niching point by PC. is 1/.0075 = 133
cm*. This accounts for the phenomenon of Regulation of ice
and the experiment of Tyndall (p. 110).
These results were quantitatively verified by Kelvin, His experi
mental results are given in Table L
Table 1 .— Depression of freezing point with titer ease of pressure.
Increase of pressure.
Depression
of
freezing point in °C.
observed.
calculated.
8.1 aim.
16.8 „
0.0580
0.1289
0.0607
0.1260
For substances which contract on solidification the melting point
will be increased by pressure. Thus for acetic acid whose melting point
is 15.5°C de Visser found experimentally that an increase of pressure
by 1 atmosphere increases the melting point bv 0,02432°C < , while
equation (50) gave 0.02421 °C. This verifies relation (50),
x.]
FIRST RELATION
263
Equation (50) will also hold in the case of fusion of metals. With
its help the change in melting point of any substance with pressure
can be easily calculated. The values so calculated, together with the
experimentally observed values, for a few metals are given in Table 2.
The agreement is fairly satisfactory.
Table 2. — Change in melting point with pressure.
Metal.
So
Cd
Pb
Bi
Melting
point
in °C.
2 M 1 .0
320.4
326.7
270.7
Latent
heat in
cal./gm.
in c.c.
per gm.
&T per
1000
atm.
(cak.)
1 t.25
13.7
5.37
12.6
0.003894' +3.34
0.00564  +5.9)
0.0039761 +8.32
0.00342 3.56
ST per
1000
atm.
(obs.)
+3.28
+6.29
+8.03
 &.55
Equation (50) may also be employed to calculate the latent heat.
For the vaporization of water at 100°C. we have the following data :
T — 373°, ?'i = 1 c.c, w 2 = 1674 c.c. per gram.
iL_ —27.12 mm. of mercury,
a I
. 373x(1674~l)x2712*lQ13xl0« ^ n ^ f
" L ~ 760 x 4185 xlO 7 '*
The accurate experiments of Henning give L = 538.7 cal. per gin.
at 100° C. which is very close to the calculated value. We can also
calculate the latent heai, of evaporation of water at various tempera
turfs from (50) if we know the values of v 2 and ^ at these tempe
ratures. The values of L so calculated are given in Table 3, together
with the observed values. The agreement is seen to be very close.
Table 3. — Latent heat of steam at different temperatures.
t°C.
dp
&=. in mm.
100
110
120
130
140
150
160
170
180
dT
of Hg,
27.12
36.10
47.16
60.60
76.67
95.66
117.7
143.4
172.7
v 2 in c.c.
1674
1211
892.6
669.0
509.1
393.1
307.3
243.0
194.3
L (calc.)
539.5
533.4
526.6
520.1
512.9
505.7
497.5
489.8
481.8
L (obs.)
Henning
03
.7
.1
525.3
518.2
510.9
503.8
496.6
489.4
482.2
"Taken from Jellinek, Lehrbuch dcr Fhysikalischcn Chetnk, Vol. 2, p. 517
(1930 edition).
264
THERMOD YNA MICS
[chap.
31. Application to a liquid film, — Equation (48) may also be
applied to the case of a liquid film. If such a film is. stretched, its
volume remaining constant, the energy equation for Lhis case is
($Q) s = du2<rdA,
for the work done by the film is 2odA, where <x is the surface tension
and dA the increase in area of the Film, The usual term pdv on the
right has disappeared since the total volume remains constant. Hence
corresponding to p and dv we have in this case 2,q and dA respec
tively. Therefore equation (49) yields
\m)t,v = " 2T \dT)»,A
and for a finite change
•q—it (£),*• (si)
For a liquid the surface tension decreases with temperature,
therefore — r is negative and 8£> is positive. Hence an amount of
heat must be supplied to the film when it is stretched in order that
its temperature may remain constant. In an adiabatic stretching the
temperature will fall by an amount
«■ ££(£■.) dA (52)
where C A is the heat capacity of the film.
32. Second Relation, — Another important application of the
thermodynamic formulae consists in their application to some adia
batic changes such as the sudden compression of a liquid or sudden
stretching of a rod. For this case let. us put x = T, y = p. Equation
(47) then reduces to
&)r — (£).• • • • ™
which means that the decrease of entropy per unit increase of
pressure during an isothermal transformation is equal to the increase
of volume per unit increase of temperature during an isobaric process.
Multiplying both sides by T we have
{£),= r (ia= r  ■ • < m >
where a is the coefficient of volume expansion at constant pressure.
From tins relation it follows that if a is positive, i.e., the substance
expands on heating,  jM is negative, and hence in this case an
amount of heat must be Laken away from the substance when the
*■]
SECOND RELATION
265
pressure is increased, in order that the temperature may remain
constant. That is, heat is generated when a substance which expands
on heating is compressed. For substances which contract on heating,,
a cooling should take place.
These conclusions were verified experimentally by Joule who
worked with fishoil and water. The liquid was contained in a vessel
closed at the top by a piston and pressure was suddenly increased
by placing weights upon the piston. The change in temperature was
measured by a thermopile. From (54) we can deduce the increase
in temperature AT produced by a sudden increase of pressure A/>
We have*
PB»
where c$ is the specific heat of the substance and v its specific volume.
We have assumed that v a /c p is independent of pressure, and then
integrated for a iinite change /\,p in p. Joule's results with water are
very interesting and are given in Table 4.
Table 4. — Increase i?j temperature of water by sudden
increase of pressure*
Ap in kg. per cm 2
Initial temp.
in °C,
AT (obs.)
AT (calc)
26.19
 0.0083
 0.0071
26 J 9
5.00
+0.0044
+0.0027
26,19
11.69
0.0205
0.0197
26.19
18.38
0.0314
0.0340
26,19
30,00
0.0541
0.0563
26.17
31.17
0.0394
0.0353
26.17
40.40
0.0450
0.04 7 U
The agreement between the observed and the calculated values
is seen to be very close. This proves the essential correctness of the
theory ; in fact, these results formed one of die earliest experimental
verifications of the second Jaw. Thus the thermodynamic theory
explains the remarkable fact that water below 4°C. cools by adiabatic
compression inspite of the fact that the internal energy is increased.
Another series of experiments consists in the adiabatic stretching
of wires. The best results were obtained by Haga. The change in
temperature of the stretched wire was measured by means of a
thermopile formed by the wire itself and another thin wire wound
round it. It will be seen that tension means a negative pressure
and hence wires of substances which expand on heating should show
a cooling when stretched adiabatically. In this case the work done
*We may directly deduce this result from equation
266
'III liRMOD YN AM ICS
dv
iCtJAP.
is not pdv but JPctt hence ?=. must be replaced by ^ where I is
the length of the wire. Hence in place of (55) we get
where /3 is the coefficient of linear expansion, ta the mass per unit
length of the wire, c p its specific heat in mechanical units and C the
heat capacity of the wire.
For a German silver wire of diameter 0.105 cm. at room tempera
ture, Haga found the mean value AT = 0,1063 for a tension of
13.05 kg. and AT = 0.1725 for 21.13 kg. tension, giving a mean
value of 0.00813 per kg. If we substitute the values o£ w, c p , /3 in
equation (56) and use the value / — 4.1S X 10 7 ergs per tab, we
get AT =0.00810 in dose agreement with the experimental value.
Indiarubber when moderately stretched has a negative expan
sion coefficient. This should show a heating effect when further
stretched adiabatieally, which is found to be true experimentally and
may even be felt by the lips.
33, Other Relations.— Besides the above two, there are other
rtant relations. Thus putting x — S* y = v in (47) we act
IS). (ft • • ■ ^
Again putting X ss s, y =/j we get
This is the third
i 9 —) = (— )
(58)
This is the fourth relation. The interpretation of these results is
left to the reader.
The above relations are known as Maxwell's four thermodynamic
relations. Besides these, there are two more relations which may
be obtained by taking p, v t Or T, j, as the pair of independent
variables. They are
GH.Oreirew. 1  • ■ «
These arc called the fifth and sixth relations and are much less used.
34. Variation of Intrinsic Energy with Volume.— From the tela
tta(£) r (§L) p . we find by substituting ^±^' for
ch
x.]
that
VARIATION OF INTRINSIC ENERGY WITH VOLUME
©rT*{fr).*
267
(61)
This equation enables us to calculate the variation of intrinsic energy
with volume.
For perfect, gases* p = y, hence ^J ^ = (Joule's law, p. 47).
BT
For gases obeying the van der Waals' law p = ,^» — y\ it can
be easily seen by substituting in (61) that
For general systems, we can transform (61) to a form which allows
du
us to calculate J from experimental results. This follows from
do
(68)
equation (IB) of page 240 where it is shown that
\dTt,r \BvlT\dTlp
= TEaPi ..... (64)
where £ = bu1k modulus of elasticity, « = coefficient of volume
expansion.
We shall compare the ratio . u : p for a typical liquid, viz.,
mercury. For mercury at 0°C and atmospheric pressure we have
as on p. 240.
(dP)  46.5 atm. em* ^C 1
1("\ = 46.5 X 273.2  1 = 12700.
Hence
du
Now — is sometimes known as the internal pressure the idea
being that the visible pressure is equal to the force per sq. cm,
with which the particles bombard a layer inside the liquid minus the
internal pressure, i.e. force/cm 2 with which they are drawn inwards
due to forces of cohesion. The above comparison shows that for
* U, V refer to a grammolecule, hence ay = ~qZ m £ e » el 'al. hi the text
both forms have been used.
TH ERMODYN AMICS
CHAP.
ordinary liquids, this force is very great but when we increase the
temperature, the ratio diminishes till it vanishes when the perfect
gas stage is reached.
35* The JouleThomson Effect. — In Chapter VI, we proved that
in the JouleThomson expansion u J pv remains constant while the
pressure changes on the two sides of the throttle valve. Let us
calculate the change in temperature due to an infinitesimal change
in pressure during this process. We have by the conditions of the
process
d (u f pv) — (65)
Since from the two laws of thermodynamics du \ pdv ss Tds, we
have from (65)
Tds + vdp = Q, ....
(66)
or
\ QW
Hence
or
r (a 8 V)/ T+T () r * +t * = °
T \dT) p ==cp; [dpfr^ " XdTlp
rlBv \
\dp )h c„
— o
(67)
 —  .
op Ik
For a perfect gas, TK J  V is easily seen to be zero and the
JouleThomson effect vanishes. The porous plug experiment therefore
provides a very decisive test of finding whether a gas is perfect or not.
For a gas obeying van der Waals* law it can be proved that
1B_T\ = 2a{Vb)*bV*RT V
[dp
(B_T\ = 2t
W hi = 1
RTV*Za{V— bf C;
idr*) 1 **"
JVOX,
(68)
ST
Hence ^— is positive, Le. when &p is negative there is cooling
as long as 2*<t„ . At T greater than ZafbR the gas becomes
heated on suffering JouleThomson expansion. The temperature
stands for the total heat u + pv or the enthalpy of the system.
x.]
CORRECTION OF GAS THERMOMETER
269
given by T. — 2a/bR is called the temperature of inversion since on
passing this temperature the JouleThomson effect changes its sign.
It will be easily seen that —=—T s approximately where T e is the
critical temperature of the gas. We have already dwelt upon this
point in Chap. IV, These results have been approximately verified
by experiments,
36. We will analyse the JouleThomson effect in another •
which perhaps gives a better insight into the mechanism of the
phenomena. We have
Now
Tds = du 4 pdv,
. , l*Z\  _/*l\ ( S (P»)\
" M3> /*" Wt \ dp IT
(69)
The first term on the right, measures deviation from Joule's law
while the second gives the deviation from Boyle's law. and the
JouleThomson effect is the resultant effect of deviations from
both these laws.
Now {') is always negative, for that part of the internal
energy which is due to molecular attractions always decreases with
decrease of volume, Le., increase of pressure. Hence due to devia
tions from Joule's law alone the JouleThomson effect will be a cooling
effect. Upon this will be superposed the effect of deviations
from Boyle's law which is a cooling effect, if
d(pv)
dp
is negative (t.«r v
&f»
before the bend in Fig, 6, p. 98) and a heating effect if —fr'
is positive.
37. Correction of Gas Thermometer, — Equation (67) can be
directly utilised for giving the absolute thermodynamic scale from
observations on an ordinary gas thermometer. For we have
'fr).*GV
*P
where all these quantities ought to be measured on the thermo
dynamic scale. In actual practice we use a gas or any other thermo
meter to measure this temperature which we may denote by 6
and c/ denotes the specific heat on this arbitrary scale. We have
_ dO _ dCL d9_ , </0
6 *~~ df d& dT cp dT'
270
Further
TIIERMODYN AM ICS
ldT\ _dT tm
\dplk d$ \dpi h
tar/," \dBfpdT'
[CHAP.
Hence (67) yields
dT
T
(!)
v+c.
\Bpn
or
(70)
The quantities occurring on Lhe righthand side can be measured on
any thermometer ; all that is necessary is that the same thermometer
should be used in all these measurements. Further these quantities
vary with temperature. As a first approximation* let us assume
them to be constant. Then on integration we get
log T = log (v + pc/) + const.
dp h
constant. This shows that the thermodynamic temperature T is
related to the volume of the gas in this complicated way and not
only as T qc v as in the case of perfect gases. Let the thermodynamic
temperatures corresponding to melting point of ice and boiling point
and Tioo respectively and the corresponding volumes
of the gas he u and v tm . Then
,, stands fior the JouleThomson effect r«— ) and
a is a
T"n
^ 1IM '
100
'j ; :''";
be the volume coefficient of expansion. Then
r. L(Hfi«l).
(71)
1 Ins gives the temperature of the melting point, of ice on the thermo
dynamic scale.
The results for a few gases have been calculated and are given
in Table 5, For hydrogen the JouleThomson cooling is 0.039°C.
per atm., Me t = 6.86 X 4.18 X 10 7 ergs per mole per °C, Mv =
22.4 X 108 c, c , per mole and a == .003661 S per D C.
* More accurately, however, we have to integrate over short intervals.
*■]
.'. T = 27313 ( I — '
EXAMPLES
039x6*86x418x10
271
! )
10 e x 224x1
275.13 (1 .00050) =273,0 degrees.
Table 5 JouleThomson correction to the gas thermometer.
Uncorrected
Volume
Mean Joule
temperature
Correction
Thermo
Gas.
coefficient
Thomson
of melting ice
term
dynamic
of expan
cooling per
fl 1
e p ' se
tempera
sion a
atm.
#o = 
0% dp
ture T
per °C.
H,
.0036613
 0.039°C.
273 J 5
0.13
273.0
.0036706
+0.208
272.44
+0.70
273,14
co 2
.0037100
+1.005
269.5
+4.44
273.9
For air the dnin are most reliable and yield the value To = 273.14°.
It will be seen that though for the various gases the melting point
on the uncorrected gas thermometer (0 O = l/« in column 4) is much
different, the melting point corrected by making use of the Joule
Thomson effect comes to about the same value, viz., 273°,
To find the correction to be applied to the gas thermometer at
other temperatures we must employ (70). Thus comparing ^ (70) and
(71), we have a .= l/v a. Then any temperature T t is easily found
if v., measure the corresponding specific volume and the Joule
Thomson effect ; for
r,+(*
+f
,de\
(72)
The eas thermometer temperature is 9i s= —  — , and the correction
° v a a
. ' aa
term is — *  r . Rut as already mentioned on paste 10 the existing
v a a dp l *
data on JouleThomson effect are not sufficient, and the corrections
are usually calculated from deviation from Boyle's law.
38. Examples. — We will now give a few examples which will
illustrate the utility of the foregoing thermodynamic formulae and
will give some practice in applying them,
1. Prove that c p  c v = ^(^) (y) = TE#v, where E h the
bulk modulus of elasticity, a the coefficient of volume expansion and
v the specific: volume.
272
THERMODYNAMICS
CHAP,
*■]
: 'I I CIFIC HEAT OF SATURATED VAPOUR
273
["»+rfr) r Tb) m+ T%) r l») t
r feh— fc)r(fr).
••• "— (Ir).(fH
[ We have ^J ^j ^) , Differentiating partially with
respect to T we get ^ = ^ , Now * p = 7" (  J f ) and therefore
U» It ' Bv5T §Tdv ~ l tar*/ J
3. Prove that
Stan From
(bp)t = ' (bt) p and P rocecd * above  ]
39. Clapeyron's deduction of the ClaasiusCIapeyron Relation.—
equation was deduced on p. 262 but the method emploved there
is open to objection because Lhe thermodynamic relations hold rigor
ously for a homogeneous substance and 'their extrapolation to dis
continuous changes is open to question. As however the relation is
important, a more rigorous deduction is given below which is due to
Ciapeyroii.
Let ABCD, EFGH (Fig. 4) represent
two consecutive isotherm ah at tempera
tures T and T + dT. From F and G
draw adiabatks meeting the second
isothermal at M and N. We can
suppose a unit mass of the substance
to be taken through the reversible Carnot
cycle FGNM, for Instance, allowing it to
expand iso thermally along FG, adiaba tic
ally along GN and compressing it along
NM isothennally and then adiabatically
along MF. The substance at F is in
the liquid state and at G in the form
Fiff. 4, Clapeyron's deduction
ui tlit ClausiusQapeyron
relation.
of vapour. The amount of heat taken during the cycle is therefore
L 4 dL at temperature T + dT. Therefose the work done during the
cycle is from equation (10), p, 216, given by
f T . m/ T+dlT\ r , _
■
: uiswiixce ueiween e\r anu jvii\, t,e. f tip tne increase u i pressure
to increase in temperature by dT, change in
volume due to evaporation or I grain of liquid and therefore equal to
yaVj where v % , tr a denote the specific volumes of the vapour and
;>!: liquid respectively. Hence the area of the cycle is
Equating the two expressions for area we get
{V % V t )dp~ i^dT,
or
(50)
_ L
Equation (50) h called the First Latent Heat equation or the
late; ion of Clapeyron.
40. Specific Heat of Saturated Vapour.— A simple expression for
the specific heal of saturated vapour may be. deduced with the help
of Fig, 3. Consider the cycle represented by the curve BFGC. The
amount of heat taken by the substance during its passage from Pa to
F is c tj dT and that during the pal s L { dL. On the other
hand Che substance gives back heat equal to c dT during the path
GC and equal to L during the path CB. c t denotes the specific heat
of the liquid in contact with vapour, and e r the s; of the
vapour in contact with liquid (specific heat of saturated vapour).
The total amount of heaE taken during th c; BFG i
c h dT 4 L f dL  c s dT  /.,
and this must, in the limit be equal to the area FGNM, which is
UT
equal to
T
as proved in the last sei
■ T+L.
^ % JTL
or
= c, ~e.
(73)
L
dT ~ r
This is called the latent heat equation of Clausius or the Second
Latent Heat, equation.
18
HMOUYNAMlCS
It may be noted that c s% is neither the specific heat at constant
p. inc heat at constant volume. Here the liquid
and die vapc rays remain in contact and therefore the vapour
always remains saturated, Both the pressure and the volume :
 that the condition of saturation is always satisfied, It is easily
seen that c s does not appreciably differ from c^ the specific heat
at co pressure, for the effect of pressure is too small bo brine
about any considerable change in the state of the liquid. c tj ea
therefore be put equal to c H . We can now calculate the value of i
from equation (73) . For water at 100°C.
= 064 cal. gmr^C" 1 ; L = 539 cal. gm." a
T = 373* : c. = 101 cal gmr^G" 1
can
c t = 101
s
0.64— = 1*07 caLgm^C'
Thus the specific heat of saturated water vapour at !00 fl C. comes
out to be a negative quantity. This is rather radoxical Tesult
ame lime perfectly true. In Chapter II we have seen
ic heat may vary from + * to  oo depending entirely
In the present case the conditio
satisfied.
i, at 10O°C, and 787.5 mm. at 101 *C.
the sp ' ater vapour
sated to 101°C. at constant pressure it
■ the condition of saturation has to be
be compressed till the pressure becomes 787.6
rhis compression generates heat, and in the case of water at 100*C.
the heat generated is so great that some of it must he withdrawn in
order that the temperature may not rise 101*. The net result
in this case is that heat must be withdrawn from and not added to
the system during the whole operation. Thus we expla
why the specific heat of saturated vapour sometimes becomes negative.
' The same idea can be expressed mathematically. We have
refers to the condition oE saturation being satisfied.
Or
e t = e.
T
r , v MM
\ST) t \dTl m .
x.
THE TRIPLK PC
275
Now for all vapours
is positive and hence c, is always less
than Cp and may even become negative.
From these considerations it will be seen that saturated steam
must become superheated by adiabatic compression, e.g,, water vapour
at 100°C. and 760 mm. pressure., when compressed suddenly to
787 mm. would be heated by 2. PC, and hence become superheated;
in other words, when the temperature of saturated steam is raised
it gives out heat. Conversely, when it is allowed to expand adiaba
tically, say, from 787 mm. to 760 mm., the temperature would fall
by 2.1 "C.j Le., to 98.9°, and hence it would be supercooled, and
partial condensation may occur. For certain vapours such as saturated
ether vapour, the work done in compressing the substance is not so
great and the specific heat is positive. These do not become super
heated by adiabatic compression.
These conclusions were experimentally verified by Him in 1862.
He allowed steam from boilers at a pressure of five atmospheres
(temperature about 152°C) to enter a long copper cylinder fitted witii
glass plates at its ends. When all the  air and ' condensed water
had been driven out, and the cylinder had attained the temperature
of the steam, the taps in the supply and the exit tubes were clos
and the vapour when viewed from the ends looked quite transparent.
The egii tube was next suddenly opened and the vapour expanded
adiabatically and a dense cloud was observed inside the cylinder. The
cloud however soon disappears as the cooled vapour rapidly absorbs
heat from the walls of the cylinder which are at 152°C. No such
condensation was observed in experiments with ether vapour.
41. The Triple Point— If we now plot the saturation value of p
against T we get a curve OA (Fig. 5) the slope at every point of
which will be given by
£,
dT T{v s v t ) »
where L t is the latent heat of vaporisation, and v g} v) denote the
specific volumes at the temperature considered.
To fix our ideas let us consider the case
of water. When the temperature is re
duced to 0°C. water freezes and we get
ice. But ice has also a definite vapour pres
sure which has been measured. The vapour
pressure curve of ice may be represented
by the line OB., the slope at every point
of which will be given by
dp L,
dT T(v t v,)
where L s is the latent heat of sublimation.
Temperature
Fig. &— The triple point
276
THLRMODYNAMICS
[CHAI\
x1
ii ie TKirLi': PO]
277
Similarly lor the phenomenon of melting, the curve OC repre
sents the relation between pressure and temperature, and the slope at
any point is given by
\ dT T(v,v t ) '
where L s is the latent heat of fusion. We have already seen on
p, 262 that for ice ?• — 133 X 10° dynes cm.  °C\ i.e., the curve
1 a r
should be almost, vertical.
For substances which contract on solidification the slope of OC
will be positive. The three lines OA, OB, OC are respectively called
vaporization line, sublimation tin •, and in the parti
cular case of water they are called steam line* hoarfrost line and ice
line respectively.
Consider the substance in the state represented by any point Q
above the line OA. It will be noticed that tile pressure of the subs
tance In this state is greater than that which will correspond to the
saturated vapour pressure at that temperature and which is given
by the intersection of OA with the ordinal a from Q. At
this iressure represented by O, therefore, die substance can
not boil at this temperature as boiling point is raised by pressure,
and mi' lore exist as liquid. Thus the region above OA xe
pres, Similarly for points below OA the correspond
too low and the substance must exist as gas.
■ substance r • as solid and below it
any point in OC due to the pressure being Larg
g to the ice line, ice will melt and therefore
presents water, and that below represents ice.
It can be easily shown that these three curves must meet in
a single point which is called the triple point. For, if the curves do
not meet at a point, let them
intersect each other forming a
small triangle ARC (Fig. 6) 
Then since the space ABC is
above AB it must represent the
liquid region, but it. is below
AC and must therefore repre
sent gas, and is also below
BC and must therefore also
represent solid. Thiis the re
gion ARC must simultaneously
represent the solid, liquid and gaseous states. Evidently it is
impossible to satisfy these mutually contradictory conditions and the
only conclusion to be drawn is that such a region doesnot exist. In
other words, all the three curves meet in a single point.
The coordinates of the triple point can be easily calculated from
the consideration that at this point, the vapour pressure of water
jtjjt. 6.— Impossible intersections of
ice, steam and hoarfrost lines.
is equal to the melting pressure of ice. The vapour pressure of
water at 0°C is 4.58 mm. and at 1 C G, 4.92 mm. Thus the vapour
ure rises by 0.34 mm. per degree, and therefore, if I is the
triple point the vapour pressure p at the triple point is given by
p = 4.58 f 0.34 t (74)
The melting pressure of ice at 0°C is 760 mm. and the change of
melting point with pressure is 0.0075°C per atm. Therefore the
melting point t at the pres.su re p will be
t =t 0.O075 
0.0075
~76CT
P
(75)
Solving (74) and (75) we get
t = 0.007455 C C and £ = 4.5824 mm,
It will be apparent that since the change in vapour pressure is too
small in comparison to the change in melting pressure, it is useless
to go through the complicated calculation given above and we can
simply assume that p = 4.58 mm. approximately (as t will be very
near 0°C) and calculate the corresponding melting temperature. Thus
( = 0.0075 :0.007455»C.
The coordinates of the triple point are therefore t — .0075°C, p
mm. At this point three phases (solid, liquid and vapour) coexist.
It was former J y supposed that the curves OA, OB are continuous.
first proved by Kirch holt that this is not so. for according to (50)
OA at 0°G =■
£ far
dT ' ~ T{v t v^
607 x4 1 8 i :• .760
— 273x21 xl0*xl0" •
= 0.337 ram. per degree.
dp , ^ , «i • \ 687x4.18x10?;:
^ for OB (subhmatipn) = ~ m ^ ^ ^ w
— 0.376 mm. per degree.
The dotted curve OA' is merely the continuation of OA, It
represents the vapour pressure of supercooled liquid. At  1°C\ we
have vapour pressure of liquid  vapour pressure of .solid = .04 mm. of
mercury. This has been verified by the expe tits of Hoi born,
Scheel and Henning.
Boohs Recommended.
1. Fermi., Thermodynamics.
2. Planck, Treatise on Thermndynam : 
H. Hoaie, Thermodynamics.
4. Smith, The Physical Principles of Thermodynamics (1952),
Chapman & Hall.
5. Epstein, TextBook of Thermodynamics, John Wiley.
CHAPTER XI
RADIATION
1. Introduction. — Even when a heated body is placed in vacuum,
it loses heat In tin is case no heat can lie lost by conduction or
convection since matter, which is absolute bote
processes, is absent. In such cases v that heat is lost by
'Radiation.' To differentiate this process from conduction, it is
enough to note that copper and v lich are ho much different
in their conducting powers, cut off radiation equally well when plated
between the hot body and die observer.
Now heat has been shown to be a form of energy, and die pro
pagation of heat by radiation consists merely in a transference of
energy. But the radiant energy in the processes of transference does
not make itself evident unless it falls on matter. When it falls on
matter and is absorbed, it is converted to heat and can be thereby
detected.
Let us now study some of the p les of radiant For
this purpose we require a soxin it as radiator and some h
ment to measure the emitted radiation.
As emitter Leslie employed a hollow i
cube filled with hot water, whose sides
coul i with different substances.
The cul" tnade that it can be rotated
about a vertical axis. Such a cube
called a] uhe and i in Fig. 1,
For die radiation
expi nployed a differ*
thermometer or a thermopile which are
described in the next section.
2. Some Simple Instruments for Measuring
Radiation. — Leslie, one of the earliest workers,
•■'irthermmneter which is
now only of historical interest. This consists
of two equal bulbs A and B (Fig. 2)
ing air and communicating with each other
through a narrow tube bent twice at right a
and containing some nonvolatile liquid like
uric acid. When the bulbs are at the
same temperature the liquid stands at the same
at in both the columns C and D, but if one
of the bulbs level of the liquid
i in the column C and rises in the column D,
By noting the difference in level of the two
columns the difference in temperature of the
bulbs can he easily calculated.
Melloni was the first to introduce thermo
piles for the measurement of radiant energy. In its original form.
Fir. 2.— Leslie dif
fitlal air tlicr
rn.:inn ter,
XI.]
PROPERTIES OF RMUANT ENERGY
279
3,— principle of the
ale,
•ofa thei consisted of a number of bismuth and antimony
Is jmned> as indicated m Fig. 3.
The left face forms the hot junction, and
the right the cold junction. The antii
bars and the bismuth bars
ihe current flot
iio t juncti o n from bismu th to an
u rmopile a 1"
i : are
arranged in the form of a cube such that
he hot junctions are at one face and the
cold junctions at the opposite face. Such
a cube is shown in Fig. 4. The near lace
iction and is cos ck. 'lire thick aote the
;„,,; tti attl, generally mica,
the differeiii layers, and the thin lines
denote the : junctions. The current
across the .soldered junc
itimony bar,
liters the next layer at B on
. Finally the cur,
m opi] thermoj
inted for use is shown in Fig. 5.
is provided with a com
rotect from stray radiatio
n..r] ,i .. :  fitallic
I the hot junction whi in use,
Other '
cribed in sections 2124.
3. Properties and Nature of Radiant
Energy.— With the aid bl these sii
apparatus, it can be demonstrated that
Radia and Li#i '
, and h;' ■ >perticB. We
mention he points of res
blance.
(i) Radium energy,
cuum, for we arc
i a hot eh < . :
lam] ' fi  is highly
: I I current passe; through
the fi lament which is not ' to
li glow, the thermal
radiation coming from tl r can be
dete nsiti+e thermopile.
lion, like Is in
straight lines, — This can be easily
for the heat coming from a flame can be easil) ci oi mterpoi
Fig. •
$r— The thermopile
(mounted).
280
RADIATTOV
[CHAF.
_..
iU
Fig, tf.— Verification of :
a screen just sufficient to prevent the light coining from ii. On replac
lie flame by a hot noi :t, no heating effect can be
detected by a thermopile. The geometrical shape oJ
can be verified by cutting out a cross from the screen and holding
close to it a piece of wood coated with paraffin wax. The wax will
melt in the shape of a cross.
(Hi) Radiation travels with the velocity of light* This follows
from the observation that the obstruction of the radiation from the
sun during a total solar eclipse is immediately accompanied by a fall
of temperature.
(iv) Radiant energy follows the law oj inverse squat; ,,, tight,
— This may be experimentally proved by the simple arrangement
shown iu Fig, 6, A is a vessel
containing hot water and having
one surface B plane and coated
with lampblack, S is a
pile. Tt will be found that if we
move the thermopile to another
position S' (say double Ehe pre
ius distance) , the del lection of
the galvanomeb :d to
it will be unchanged
area of die surface of B from
which radiation can reach the
d four time amount of radiation
intensity of radiation un
reduced' to ii;
can be easily
g two
parabolic mirror
and B at some disl
once apart with their
axes in the same line
(Fig. 7) . A luminous
source placed at the
i produces
an image at the !
of B. If we replace
the luminous source
by a hot nonluminous
one, say a Leslie cube,
and put a piece of tinder at the other focus, the tinder is ignited.
This shows that heat and light travel along the same path and hence
obey identical, laws. In fact all the laws ""and results of geometrical
• to it,
Fig. 7. — Verification of laws of reflection for
radia
XI.
OF RADIANT ENERGY AND LIGHT
281
■') Radiant enerp rum
may be obtained by retraction through a prism (sec. 29),
((■ii) Thermal radiation exhibits the phenomena of interference
and d: ■■■'. — Diffraction can be easily observed with the he
a concave grating or by employing a ti ission grating made of a
number of equidistant parallel wi
(vii radiation can also be polarised in the same way
as light by transmission through a tourmaline plate. This can be
. by the following experiment : — Arrange that Kght from an in
candescent lamp traverses two similar plates or tourmaline. On
rotating one of the plates the image will be found to vanish. Replace
the lamp by a Leslie cube. Now a thermopile placed in the position
occupied by the image will record no deflection ; if however one of
the plates be rotated a deflection will be observed.
4. Identify of Radiant Energy and Light — Continuity of Spectrum. —
All these observations lead us to the conclusion that radiant energy
is identical with light. As we shall see presently, Radiation or
Radiant Energy is the general and more expressive term,
rather visible light.) being only a kind of radiant energy which has the
distinctive power of affecting the retina of the human eye, and thus
producing the sensation of colour. But like other kinds of radiant
energy light is also converted to heat when it is absorbed by mi
The identity of light and radiant energy can further be seen from
the ft >1 ' owi 1 1 g v. Kpe rimen ts : — ■
When we prod 1 ctrum of the sun by means of a prism T
it ts terminated on one side by the red, and on the other side by
the violet. But it can be easily seen that these limits are only
apparent, and are due to the fact that the human eye is a very
imperfect instrument for the detection of radiation, W. Herschel*
n'lSOO) placed a blackened thermometer bulb in the invisible
beyond the red. and found that the thermometer recorded a rise in
temperature. The rise in temperature was observed also when the
mometer was placed in the visible legion. Thus W. Herschel
discovered the infrared part of the solar spectrum, and showed that
it was continuous, with the visible spectrum.
The source of radiation in this case is the sun, which may be
regarded as an intensely hot body. But it may be substituted hj
••uffidently hot substance, say a piece of burning coal ; the posi
tive crater of the arc, or a glowing platinum wire, Only in th ■
the intensity of radiation is not great, mid the spectrum does
not extend so far towards the violet.
Hi) n of light by .~A very simple experiment
suffice to bring out the point which has been just mentioned.
Suppose we take a piece of blackened platinum wire and pass through
n Herachel (1738—1822) was bom at Hanover but settled in England.
I worked as a musician and later became ski astronomer. He discovered
• planet Uranus in 1781 and later discovered! the infrared radiations.
282
RADIATION
[CHAP.
XI.
EI.ECTROMAGNETIC WA
it a continuously increasing current. The function of the current
is simply to heat the wire. We find that it becomes warm, and
sends out radiant energy, If a thermopile be held near it, t*M
nometer connected to it will show a deflection* When a slightly
stronger current is passed the wire begins to glow with a dull red
light. This shows that the wire is just emitting red radiation o£
sufficient intensity to affect the human eye. Accurate observations!
show that this takes place at about 525 ,J C (Draper point) . With
asing temperature, the colour of the emitted radiation changes
from dull red to cherry red (900 D C), to orange red (1100°C), to
yellow (125G*C), until at about I600°C it * es white. Thus the
temperature of a luminous body can be roughly estimated visually
from its colour. L Such a colour scale of temperature is given in the
chart on pp. 2526, This shows that si waves are emitted by a
d body in sufficient intensity only with ini
Vice versa, we may argue that when the temperature of the wire
is below the Draper point, it is emitting longer waves than the
red, but these waves can be detected only by their heating effect.
Radiant Energy or Radiation is thus a more general name for
Light. It can he of any wavelength from to co as illustrated
ie chart i s 283284. light forms less than 8/4 ths of an
■ "
Ex m of the Chart. — We here the mic
1 along the vertical line. Thus (5) on
he wavelength is 10 B cm,, (—2) inda
hart visible light extends from
;l4x lHon. (violet). The infr;
W. Hei .tension towards the
tb side is chiefly due to die reststrahlen method of Rubens
and a are detected chiefly b thermopile,
for or plates are not sensitive beyond the green.
romatic plates may be used, but they are also
i p. mi beyond 8000 A. U. Beyond this plates dipped in
or neocyanine may be used up to 10000 A. 17. Recently
photographic plates sensitive up to 20000 A. U. have been placed on
the market.
The Hertzian waves are produced fay purely electromagnetic
ids (due to oscillation of current in an indue ►acity cir
cuit) and were first discovered by Hertz in 1887. Marconi ap
i for wireless transmission (1894). Waves used for this put
are generally 10 — 500 metres in length, though waves of about. 10
length have been used in the past. These long electromagnetic
from the light waves only in their wavelength, and
i :4s have been made to get shorter and shorter waves by purely
electromagnetic methods so that one can pass continuously to the
f According to a recent experiment by Bs . persons who have r
their eyes in the dark carefully, can detect radiation even from a h
to about 400 °C.
ELECTROMAGNETIC WAVES
Wavelength
Log
Radiation
Generation
Detection anc
Analysis
Discoverer
cms
Siegbahn
unit
11
Hard
Y rays
Reactions
Paii 
production
effect
10 ^
•0GA.U
'".
yrays
>
Radioactive
Disintegration
Photo electric
■:,hcd
10*'
Angstrom
unit
] Xrays.
Hard
~ 8 ' Medium
Bomhardment
targets by
electrons
Cry*
Photography
charal
jttiintgenue&J
Laue mm
7
/ Soft
i
i
Compton[i9£5!
ThitmudiUtefl
136 A,U. • " 6 *n
10XKP5 •jL t ^ k
Vacuum Spark
Spark. Arc
KLame
Photogr
aphy .
Pri: ma
. a
: •:: ■.
._: 9 
1
IS
~ .'
11
8
Millikan ctsi9>
&
Lyman ami
Schumann
■::■:. J:
RitterdMfij
NEWTON
HerSchelO&iO)
3
1 Infrared
Heating
Wire gratings
& Tharroo
cooples
1 mm 
2'
Residual ray
Method
Rubens &
Nicholsassa)
'4 mm ■
4mm
J
1
0.
J Hertzian
J waves
1 SI Dl
Spark gap
Coherer
Wood ana
Nichols &
Tears (wao
BoBeoffla J
mm
284
RADIATION
CHAP.
XI.]
FUNDAMENTAL RADIATION l'UG<
ELECTROMAGNETIC WAVES *»w
Wavelength
b.
Radiation
Generation
Detection and
Analysis
Discoveror
cm
sr
Spark gap
discharge
+i
Hertzian
waves
Triode valve
Radar
(Micro
1 metre
+2
Oscillator
wave
Technique)
Hertz 1\m
3
Shert'Wive
Wireless
Methods
100 metres
4
Herteian
10 «M
B
WH
7
Ne upper
limit
B
reached are
due to
between
lone infrared rays. The: limits
Lebedew, Lampa Sir J. C. Bose (4 mm.) and the
dect^magneti/ waves and beat waves has been completely fan*
hv the Nichols and Tear, Arcadiewa, and others. Duiii^
World War II the technique ol prodm tion of waves of lengtn ironi a
few centimetres lias undergone a . te revolution
. . ,>f magnetron and klystron tubes used toi Rat
The ultraviolet part (rays shorter than the last visible = 3800)
was d i sc IT. Ritter in 1802. lie round I .notogr aphic
plate was affected even beyond the visible limit. < f*«
usually give lines as far as A = 3400, after which either uvioi glass
flimit A = 2800) or quartz prisms should be used, At about A — <>
the gelatine used in the photographic plates begins to abs ,
Schumann was the first to prepare plates without gelatine and open
up what is called the Schumann region. Below A== I860 A. U. qua
begins to absorb heavily, and Schumann therefore used fluontr
Bpectroscopea. Beyond A = 1400, air absorbs heavily, and the whole
ope. ! producing light and photographing the spectrum must be
done in he pioneer in this field was 1 h Lyman o Han i
W hc ; le to photograph Hues as far as A = 600. In this region
, ra1 ; i be used for producing the spectrum.
X f ; shown by M. Lane in 1912 to be light waves of
remely short wavelength 10— « cm., w, about 1000 times shorter
than ,, light. He used a crystal as diffraction grating, lhe
shortest. Xrays measured by the crystal method has the wavelength
I U. /rays, obtained in radioactive disintegration of the nuclei
oL atoms, are still shorter, viz., from HH> to 10«> cm, in wavelength.
The gap between the Lyman region (600 A. U.) and the soft
Xray region (aboul 20 A. U.) lias been gradually by the works
of Millikan and Bowen, Compton, Thibaud and others. Mlllikan and
Rowen used ordinary vacuum spectrographs with diffraction g
Their source of light was condensed vacuum spark. For ' wavelet i
in this region, crystal gratings have then spaces too small, while tfo
ruled gratings have their spaces too large. Conipton instituted
method of obtaining spectra in this region with the aid o
grating at glancing angle (see M. X. Saba and X. K. Saha, Treatise
.Modern Physics, Vol. I, p. 273).
Since radiant heat and light are identical, all the laws and
theorems of Optics and Spectroscopy can be applied i udy
of m. Rut in this chapter, we shall deal with the subject
only as far as it is connected with heat. We shall first enter into
a preliminary discussion regarding the passage of radiation through
matter,
5, FnndamentaJ Radiation Procewe*.— Every hot body emits r;
tion from its surface which depends upon the nature of the surJ
its size and its temperature. This is known as emisai
When the emitted radiation falls on matter, a part i\ reflet
286
RADIATION
[CHAP.
and transmission are connected by the relation absorption
where r — fraction of total energy reflected
a — " " » » absorbed
i . _
— M « » « transmitted.
all ^ 'f CB n ° r t S nsm ^ the % ht ««<* falf/onTTC ,'
all, and hence appears black. But till uerfectlv hkrfr ui  "
6. Theory of Exchanges (Prevost— 17ft2l _1>»tW t^ i*oo n j
regarding emission of radiant enefgy Ze T^ „ ui f ^w*?
t a hlocfc of ice, we feel a senJrinn S ? ™ n i Wwn * e stLind
being at about W lose mLI ~ v C .V M beCTOM om b0 <*3
* Ice, which is a a SS w5. ? iadaiVm . than il rece "
phenomena. q * al dnd ma >' ** a PP lie ^ to all si,,
V>teZ^ t ^ ( ?£p^ d ™ «*! ^en it ^ at the absolute
becomes necessar tteate hnL it I  ■ ^perature, it
temperature and the nSM^Ste Tth£ 2?" ^ A
confine our attention to the *>n> t *t ln ™ S€Ctlon we shall
stance and the tm'roundin^ V,\ *«&****, both of the sub
The experiments can be ^^p^SeVV S^
XI.
LAWS OF COOLING
bulbed mercury thermometer and enclosing the bulb in an evaa
flask, the walls of which are blackened inside and are further ma 
tained at a fixed temperature by means of a suitable bath. The
rate of fall of temperature of the thermometer gives the rate of
cooling. It was found that if we plot the rate of cooling  ^ as
against die excess of temperature as abscissa, the curve is a
: ight line, or in other words,
£ H04) «
■ B n is the temperature of the enclosur he rate of cooling
is proportional to the excess of temperature of the substance above
the surroundings. This is Newton's law of cooling. The law holds
when the temperature difference is not large. Even the earlier
experiments showed considerable deviations from the law when the
temperature difference exceeded 40 a C.
In order to find out a law which wiH hold for all differences ol
temp;: of experiments were performed by
Dulong and Petit who found that their results were given by a formula
of the form
_d0
' tit
= k{aO —a&o),
(2)
where > instant and k depends upon the nature oil the emitting
e, and 0,6 O denote respectively the temperatures of the emitting
. jd the surroundings.
.Stefan* showed later that the results of Dulong and Petit
be represented by the equation
£• <*•*».
(3)
where a is a universal constant, and T and T the absolute tempera
tares oi the body and its surroundings ; or in other words, the emissive
t of a substance varies as the fourth power of the ah
erature. This is Stefan's law and can be deduced frort] theo
. 1 considerations (vide sec, 26). It is therefore the correct law
of cooling for black bodies,
Newton's law of cooling can be easily deduced from Stef;
law when the temperature difference is small. Thus if the body
at temperature T f 87* is placed in an enclosure at temperature T°K,
the rate of loss of heat, per second by unit area of the body is
a (T+8T)*aT*
 „T* / 1+"^ ) ' ^r 1 = ±vTHT
neglecting higher powers of ST. It is thus proportional Co the tem
t.ure difference 87'.
♦Josef Stefan (1835—1893) was Professor oi Physics in Vienna. He dis
covered the law known by his tuutie,
288
KADIATIOK
[CHAP,
Generally for rough work in the laboratory ! law of
cooling is employed Eor correcting for heat loss even when the body
suspended in vacuum. As shown in Chap* VIII, § 30, if
cooling takes place by natural convection in air, Newton's law
not hold. In accurate work tct rate of cooling should be
irved and not computed.
Exercise*— A, body cools from 50°G to 40°C. in 5 minutes when
trroundings are maintained at 20°C. What, will he its tempera
ture after a further 5 minutes? Assume Newton's law of cooling
to hold.
Nov, for .: ^ 0, = 50
t=W t 9 = X
.*, C  lag, 30 j £ = 
,. log (020) =kt + C.
5,
(to be determined) .
1 . 20
5 ' ''" 30
•'■ log4 "itr = 2 logj M whcncc 6 *=M3*s]
8. Emissive Power of Different Substances — Preliminary Experiments.
—Having n experimental study of how the emissive power
with i let us find how the
with the nature of the surface ig body, is pur
Leslie : ted with i i while
I with the sul to be
n both 1 to fall
itio of
led by Lite thermopile in Lh gives,
In this
missivitii mined,
Lesl
9. Reflecting Power. — T.I tig power of difl mb
found by allowing the radiation from a
Leslie cube to n Be in surface and the refle<
, Next the thermopile is placed at the
surface. The ratio of ns in
I Meeting powi rbis
^e wi th I radiation employed, and as earlier workers did not.
employ monochromatic radiatit i lults are m i ach value.
10. Diathermancy. — Diachermanq with regard to heat is
analogous to transparency in the case of light. Subsi which
allow radiation to pass through them are called diatk
those [o not, are "called atk\ us. The early experiments
in this di performed by Melloni and I. Direct
radiation from a Leslie cube fell on a thermopile produ deflec
tion ; next the experimental substance was introduced and the deflec
tion observed. The ratio of the two deflections gives the diathermancy
XT.
RELATIONS BETWEEN THE DIFFERENT RADIATION QUANTITIES
289
or transmission coefficient of the substance. Solids, liquids and gases
were treated in this way. The best diathermanOi are rock
salt {Nad), sylvine (KC1) , quartz, fluorite mid certain other crystals,
diathermanous, but water vapour and carbon
dioxide show marked absorption.
11. Absorption. — For studying the absor] lids Melloni
took a copper disc and coated one face with lampblack and the other
Tare with the experimental substance. The coppei dm was placed
v,„ the L' abe and the thermopile, the Ian black surface
the thermopile. The plate will assume mperature
T. ' al radiated per unit area of the plate
H= (E + B)T.
where E and E denote the emissivity of lampblack and of the
substance. Now this must equal the radiation which it absorbs.
The thermopile reading gi*es ET and therefore E. Knowing E and
/,;■■ we g e i //. th< ii.sorbed. In this way the absorptive power
of different si es could be compared. For gases a sensitive
arrangement ed,
terms diathermanous or athermanous lark in scientific
precision. Every substance ought to be defined, so far as its trans
mitthij sorting properties are concerned, with respect to a parti
cular wavelength. In the above experiments monochromatic radia
were not employed. The whole subject is now studied as a
branch of physical optics under the head 'dispersion and reflection'
with which it Is intimately connected.
12. Relations existing between the different Radiation Quantities.—
The foregoing experiments however show that at the same tempera
ture a lampblack surface emits the maximum energy while a polished
surface emits very little energy. It was further found that radiating
and absorbing powers vary together; that good radiators are good
absorl I pocr reflectors, while poor radiators are poor absorbers.
Lampbin aque to radiation but allows radiation of very long
wavelength to pass through it.
A very simple experiment devised by Ritchie demonstrates
vividly the relation between emissive and' absorptive powers of a
body/ A Leslie cube AB, which is a hollow
metallic vessel and ran be filled with liquid
at any temperature, is placed between the
two bulbs of a different; aeter
(Fig. 8). The face A of the cube and the
bulb D are coated with lampblack while the
face B and the bulb C can be coated with a
layer of the substance whose emissivity is to
be' investigated, my powdered cinnabar. By
filling the cube with a hot liquid the index
not found to move, a ace the
amounts of energy received by C and D are equal. If e denotes the
19
C
6. — Ritchie's experi
ment
RADIATION
[CHAP.
►f heat emitted by the substance and a its absorptive power,
lenote the corresponding quantiti. : ampblack,
i
e
E
a.
w
Now <?/£ may he called the coefficient of emission o£ the sub
atancej and hence the relation shows that the coefficient of emission
ual to the absorptive power. It will be seen that the effect of
temperature upon the coefficient of emission has been neglected here.
Nevertheless the experiment shows at least qualitatively that the
coefficient:? of emission and absorption vary in the same manner from
one substance to another.
These ideas were further developed and made more precise by
Kirchhoff and Balfour Stewart. Before proceeding further it is how
ever necessary to define the concepts used in Radiation with more
rigour. We now proceed to do this in the next section.
13. Fundamental Definitions. — If a body is heated it radiates
Uons its surface in all directions, which comprises waves of all
The nature of radiation depends on the physical properties
of the boi denote by e\dk the amount of radiation measured
js emitted normally per unit area per unit solid angle per second
within the wavelengths A and \\d\. We shall call e ?i the emissive
of Che body. arly if dQ\ be the amount of radiant energy
a the body in the form of radiation (A to A + dk) and a
absorbed by the body and converted
is called the absorptive power of the body for these
to Af dX). For black bodies a\ =1 for all wavelengths,
r other substances a. depends on the physical nature of the body.
14. Kirchhoff's Law.— In sec. IS « Chat emission and
i vary together. In 1859, Kirchhoff* deduced an important
which may be stated as follows : —
%iven temperm ' ratio of the emissive power to the
''live power is the saute for all substances and is equal to the
>er of a perfectly black body.
Though this law was first recognised bv Balfour Stewart,
the first to deduce it from therm odynaniical principles,
apply it in all directions. It is therefore usually known as
KirehhorTs law.
We have considered here the total emission regardless of wave
Li but the same relation holds for each wavelength separately.
radiations of the same wavelength and the same tempera
* Gustav Robert Kirchhoff (1824 — 1887), bora at Kfioigsberg; became Pro
fessor of Physics at E and Heidelberg, He h noted for his discoveries in
spectrum analysis and the radiation law bearing his name.
XI.]
APPLICATION OF KIRCHIJOFF S LAW
291
turcj the ratio of the emissive and absorptive powers for all bodies
is the same and equal to tl live power of a perfectly black body.
This also holds for each planepolarised component of any ray.
Kirchhoff's own proof of the law is however very complicated
and has. so I fallen into disuse. We shall give in sec. 19 a much
simplified proof of the law. But before doing so let us consider its
applications.
15. Applications of Kirchhoff's Law.— The law embodies two
distinct relations, a qualitative and a quantitative one* Qualitatively,
it implies that if a body is capable of emitting certain radiations it
will absorb them when they fall on it. Quantitatively, it signifies that
the ratio is the same for all bodies.
Various experimental proofs and observations may be cited in
support of die qualitative relation. If a piece of decorated china is
heated in a furnace to about 1000°C and then taken out suddenlv in
a dark room, the decorations appear much brighter than the white
china, because these being better absorbers, emit also much greater
light, If we take a polished metal ball and have a black spot on it
by coating it with platinum black., then on heating die ball to about
I000°C and suddenly taking it out in a dark room, it will be found
that the black spot is shining much more brilliantly than the polished
surface. Again take die case of a coloured glass. We know that
green glass looks green because it absorbs red light strongly and
reflects the green (red and green being complementary colours) .
Hence when a piece of green glass is heated in a furnace and then
taken out, it is found to glow with a red light. Similarly a piece of
red glass is found to glow with green light. A more decisive example
illustrating selective action is that of erbium oxide, didyrnium oxide,
etc., which when heated emit certain bright bands in addition to die
continuous spectrum. If now a solution of these oxides is made and
continuous light, is passed through it, die very same bands appear in
absorption.
1$, Application to Astrophysics. — B e s i d e s these applications
Kirchhoff's law was in a sense responsible for the birth of two entirely
new branches of science, viz., Astrophysics (physics of the sun and
the stars) and Spectroscopy. We shall recount here briefly how these
developments grew out of Kirchhoff's law, Newton had shown in
1680 that the sunlight can be decomposed by means of a prism into
die seven colours of the rainbow, but Fraunhofer who repeated the
experiment in 1901 with better instruments, found to his surprise,
that the spectrum was not continuous, but crossed by dark lines.
Their number is at present known to be 20000, but Fraunhofer noticed
about 500 of them, and denoted the more prominent bands by the
letters of the alphabet : — A, B, G, Such a solar absorption
spectrum is shown in Fig, 9 (C) togeiher with a continuous prismatic
(A) and continuous grating spectrum (B).
T
RADIATION
CHAP,
Frau nhofer never understood how these dark lines originated,
neither did any oi Jus contemporaries. But he realized ilu i: great
red them and catalogued their wavelengths. He
examined the light from stai . and showed that then .p
also crossed by dark lines just as tn the case of the sun. A tj
stellar spectrum is shown in Fig. 9 (E) .
In : antime, however, other sources of light, were examined
by the spectres id some knowledge was obtained of their spectra
and that a glowing solid gives a continuous spectrum, hut
a flame tinged with NaCl pair of intense yellow Hues
dark background. It was also found that, if an i discharge was
I through a glass tube containing gas at low pressure, a large
number of emission lines were obtained on a dark background, For
hydrogen these are shown in Fig, 9 (D) .
But still the dark lines of Fraunhofer remained unexplained.
Some physicists, notably Ffeeau, o I that if the spectrum or
the sun >  Id with the spectrum from a sodium
flame, the yellow lines appear in the same place as the Dband of
the FraunhoFej rum. Similar is the case with the hydrogen
rum. in Fig. 9(C) ai I we see the emission spectrum of
hydro by side with the Fraunhofer spectrum. It is clearly
m some of the dark lines in the latter occur in the same place
bright lines in the former.
Th Iven by KirchhoS not only completely solved
; : rreacning and extremely fruitful in its con
ned that the central body of the sun consists
ich emits a continuous spectrum without I
i lighl has to pass through a cooler atmosphere sur
mass. In this atmosphere all the elements like
c etc, :n ]i, in the gaseous form in addition to
' :. ::tc.
W: : ri how the early workers studied the phenomenon
of radiation with simple apparatus and arrived at some very
general laws. Every substance when heated emits radiation, Le tJ
light. Every substance has again got the power of absorbing light.
Kirdihoif arrived at the same law from thermodynamic reasoning
19) and applied i( to explain the dark lines. Sodium can
the DIines when it is excited; hence when white light falls on it, it,
can abs ; ' also the same light, and allows other light to pass through
it unmolested. The gases in the outer cooler mantle round th
therefore deprive the continuous spectrum From the central mass, of
the lines they then ran emit, and give rise to the black lines.
The Dbands therefore prove that there is sodium in the sun's at
mosphere. Similarly, the other dark lines testify to the presen
their respective elements in the atmosphere of the sun.
The correctness of Kirch hoi i matJon. is seen farther from
flash spectrum results. We have supposed that 'the atmosphere of
To face p. 292
CONTINUOUS PRISMATIC
CONTINUOUS GRATING
SOLAR ABSORPTION GRATING
FED C B B a
A
(oTT
rafw , , , 
(D)l
HYDROGEN BRIGHT LINE
SPECTRUM OF ^AURIGA (G r , TYPE)
(E)
■
COMPARISON OF FLASH SPECTRUM
(1905) WITH ROWLAND ATLAS
(Flash photograph enlarged six times and Atlas reduced five limes).
(F)
m ii ii Aiiiiiiiliiiiiiiiiiiiiiii
Fig. 9
Figs. (A), (B). {€), (D) have been reproduced from Know] ton, Physic
flege Students, (E) from MullerPouilicts. Lehrbuch der Physik, Vol. V,
part 2, and (F) from Handbuch der Astrophysik, Vol. IV.
I
XI,
APPLICATION OF THERMODYNAMICS TO RADIATION
the sub contains Na. Now if we could observe the spectrum of the
atmosphere apart from that of the central glowing core (the photo
sphere}, tlu: line uid appear I We cannot ordinarily do so
because the solar atmosphere is so thin that we cannot cover up the
disc properly, leaving the atmosphere bare for our observation ;
further the scattered skylight completely obscures the spectrum of
the pari outside the dis during a total solar eclipse, the
solar disc i ■ completely coveted for a short time by the disc of the
moon, and skylight is also reduced by the moon's shadow. To anyone
observing the sun through a spectroscope, the solar atmosphere will
be laid bare al the time of totality and the dark lines will flash out
as bright lines. This was Ei iii d actually to be the case by Young of
Princeton in 1872, Fig. 9 (F) shows a Hash spectrum of 1905 eclipse
which is placed side by side with the Rowland Atlas of Fraunhofer
spectrum for the sake of comparison. The lines of the flash spectrum
are thus found to be due to the elements in the sun's atmosphere.
But Kii Hi ho ]"s disco voi}' is of much more farreaehing import
ance than the mere success in explaining the. Fraunhofer bands would
indicate. It clearly asserted for the first time, that ev& rent
type of when it is properly excited, emits light of defcni
wavelength which is characteristic of the atom. Just as a man is
known by his voice, ot a musical instrument by the quality of its
note, so each atom can be recognised by the particular lines it emits.
Thus was bom the subject of Spectrum Analysis, which aims at
identifying elements b) their characteristic lines, and forty new de
nts 'were added with its aid to the list already known. The
different aton rded as so many different types of instruments,
each capable of producing its own characteristic aetherial music.
APPLICATION OF THERMODYNAMICS TO RADIATION
17. Temperature Radiation. — Since material substances at alj
temperatures are found to emit radiation, it becomes possible to
apply the laws of thermodynamics to the problems of temperature
radiation. The expression "Temperature Radiation" should I
clearly understood, for matter can be made to emit radiant energy
in many way. othei than by heating, e.g., by passing an I '• trical dis
charge through it when in a gaseous state, by phosphorescence, fluores
cence, or by chemical action as in flames. But Kbchh oil's law holds
only for temperature radiation. For radiation produced by other
methods the law cannot be applied,!
18. Exchange of Energy between Radiation and Matter in a Hollow
Enclosure. — Let us suppose that there is an enclosure with its walls,
which are impervious to external radiation, maintained at a cons
tant temperature. We shall study what takes place when we place
substances having widely different physical properties within this
enclosure.
In the first place the whole space is fdled with radiation which
is being emitted by the walls. This radiation arises out of the heat
294
RADIATION
* M??^ b ^ zi "^ «W««W*y squire the same temperature
the walk* ihu can be proved by the method of nducfja
***. .tor suppose that inV equilibrium state the tem
evSkLJi 11 a £a™°t engine w*T ^ used to a heat
ESTSS T, ' W ? lls t0 A unU1 A J,ad die samc temperature as
the walla. During the process a certain amount of heat will
SmESJ 1 ^  But ^ in A , would be reduced automatically i
Si r /^J*™"" ;f according to our assumption, this is the
.table state o/ affairs. Hence again the difference of at ure can
he l hcdv J^ f* C0nVei ' SiOn ° f hCat f ° W ° rL We • **™
wk ' 7** ° 38SU ^ a temperature different from that of the
g. we have at our disposal a means of continuous!, . ting
the heat of a single body to work without mamtaininj
conclude tfut ri materia] bodk:  J n the enclosure woi
mately assume the temperature of the walls.
fa vi^V'Vr' fPPosf that a body of hti [position
£P 'Inn the enclosure. Then the different parts of iL'body"
toe different emissive and absorptive . The total euro
ab> " * equal to tile energ , S\iSfS
temperature reman, llt [n the equilibrium , Z f t j£
T" »m PrTvost's
2J « Allows the
*  « ****** is changed, And as the d'fS
l" 1 ' Cerent coefficients of , ,,,. rhl ca"i
™ h mica! in qua, ..,., L cpiab
of the enclosure or of any body plao [e it. T.ei L £
we have two enclosures A and B having walls of differs X w
^n?^ion u " fca ; ing f, h  cach *** tht " ?*s*
?h™ i?V xt of L WaveIe "St h lymg between A and A + o'X to pass
through it. Now the walls of both A and B are mainlined at Z
™ jure T. Tf in the steady state the intenritv of radt at o^
ms thrown ?£ «f ater * A than in B. some fadiation will
Thi Iv« and decrease in A. The screen is then closed
The excess radiation will be absorbed by the walls of R and raise to
temperature to r, while the temperature of thesis of TwTfaTL
deduction of kirciuioff's law
Now a Carnot engine may be worked between these two temperatures
ding :l certain amount of work and lowering thereby ipe
■ '■;' li and increasing that of A till the two are brought to the
irature. The process can be repeated and thus we
Duntol 'work while the entire system : cm
colder. Thus in effect we are getting work indefinite!) by
usii Meat of a single body. This is Impossible by the second
lynamics. Hence the intensity of radiatioi
the same in the two enclosures, i.e., the quality
. of radiation depends only upon the temperature and on
ii i thing else.
If we now place a black body inside the enclosure, it will emit
.in energy of the same quality and intensity as it absorbs. Hence it
follows that the radiation inside the enclosure is identical in ev
the radiation emitted by a black body at the same M
perature as the walls ol the enclosure. These concha were
rchhofE in I
19. Deduction of Kirchhoifs Law.— Suppose we 5
some sub ive and
of wavelength lying between A and
iid ax wly. We have alread a in the
don that the amount of radiation dQ falling on the sub
die wall i depend upon the nature < hape
of the walls. Of this a portion ot A dQ is I ■ nrbed b)
remainder (1 a A ) dQ, is reflected or transmitted. Further
the su radiation equal to e^dX by virtue
perature. Equating the energy absorbed to the energy emi
■
e x d\ = ax<i(l
In the case of a perfectly black body of emissivity have
(since a\ = I) ,
Combining (5) and (6) we get
 F
: — r. ;
• (6)
(7)
■;■ ratur the ratio of the emissive power to the dbsarp
ibstance is constant and equal to the emissive p
of a perfectly black body. This is the thermodynamic proof oi
Kirchhoftfs law. We have proved the law here for bodies inside the
enclosure. Now since tire emissive and absorptive powei I only
upon the physical nature of the body and ijui upon its sin i
it follows that the law will hold for all bodies under all conditions Eot
pure temperature radiation.
II ■
29G
KA IU ATI ON
[CHAP.
20. The Black Body,— The considerations put forward in sec.
enable us to design a perfectly black body lor experimental purposes.
seen that it an enclosure be maintained at a constant tern
are it becomes filled with radiation characteristic of a perfectly
black body. If we now make a small hole in the wall and examine
the radiation coming out of it, this diffuse radiation will be identical
with radiation from, a perfectly black emissive surface. The smaller
the hole, the more completely black the emitted radiation is. Thus
a correction has to be applied for the lack of blackness due to the
finite size of the hole. This h due to the fact that some of the
radiation coming from the wall is able to escape out and the state of
thermodynamic equilibrium as postulated in section IS does not hold.
This is almost completely avoided in the particular type of black body
due to Fery (Fig. 11, p,' 297) . So we see that the uniformly heated
enclosure behaves as a olack body as regards emission and if we make
a small hole in it, the radiation 'coming out of it will be very nearly
blackbody radiation.
Again such an enclosure behaves as a perfectly black body*
towards incident radiation also. For anv Tay passing 1 into the hole
will be reflected internally within the enclosure and will be unable to
escape outside, This may be further improved by blackening the
Hence the enclosure is a perfect absorber and behaves as a
ectly black body.
Though Kirchhoff h cd in 1S5S that the radiation inside
3%ffi g ^a s f^,a3gESBSafefli j ^
Rheostat
Fig. 10.— Black body of Wien.
a uniformly heated enclosure is perfectly black it was long afterwards
in 1895 that Wien and Lunmier utilised this conception to obtain a
black body for experimental purposes, m
The black body of Wien consists of a hollow cylindrical metal! t
chamber C (Fig, 10) blackened] inside and made of brass or platinum,
* Kirchhoff defines a black body as one "which has the property oi allowing
all incident rays t enter without refketii u ;iii * them t:> leave again.
See Planck, Wormcstrnhhing.
 The walls however need not be black. Blackening merely enables the
equilibrium state tu be reached quickly.
XI,] RADIOME1 297
depending upon the temperature that it has to stand. The cylinder
ated by an electric current passing through thin platinum foil as
indicated bv thick dashes. The radiation then passes throng
number of limiting diaphragms and emerges out of the hole O. The
cylinder is. surrounded by concentric porcelain tubes The tempera
ture of the black body is given by the thernioelement T. This is
the type of black body now commonly used.
Another type due to Fery is shown in Fig. 11. Note the conical
projection P opposite the hole O. This is to avoid direct radiation
from the surface opposite the hole which would otherwise make the
body not perfectly black,
A striking property of such an enclosure is that if we place any
substance inside it, the radiation
emitted from it is also black and in
dependent of the nature of the body.
Thus all bodies inside the enclos
lose their distinctive properties. For
a mirror placed inside will reflect the
black radiation from the wall and
hence the emitted radiation is black.
Any substance if it absorbs any radia
tion transmitted from behind must
emit the same radiation in conse
quence of Kirchhoff *s law and the
total radiation leaving it must be
come identical with that from the walls i.e., of a black body.
Fi.cr. 11 — Black body of Fery.
RADIOMETERS
21. Sensitiveness of the Thermopile. — We shall now describe the
instruments which are used for the measurement, of radiation. The
differential air thermometer, which was employed by Leslie and the
early workers, has already been described in sec. 2, but is now only
of historical interest.
Among the modern instruments, the thermopile has been des
cribed in sec, 2,. Special care however has to be taken in order to
make it sensitive as we have sometimes to measure very small amounts
of energy. In the design of thermopiles, the following 1 considerations
have to be borne in mind : — (1) metals used should give large thermo
F..M.F. ; (2) junctions should be as thin as possible ; (3) connecting
wires should be thin so as to minimise the loss of heat, by conduction ;
(4) the junction should be coated with lampblack so that all the heat
falling on it may be absorbed ; (5) it should be mounted in vacuum,
so that there is no loss of heat by convection, and the deflections
remain steady ; a rocksalt window is provided to let in the incident
radiru
The sensitiveness also depends upon the number of thcrmo
junctions but this cannot be increased indefinitely as the external
298
RADIATION
CHAP.
resistance will increase. The best procedure is to have the piles as
light as possible and to choose a number so that the total resistance
of the thermopile is equal to the galvanometer resistance.
The galvanometer to which the thermopile is connected should
be o£ low resistance type with high voltage sensitiveness. For ordinary
work the .suspended coil type galvanometers are generally employed
but are not sufficiently sensitive on account of their high resistance.
For accurate work the suspended magnet (astatic) type of galvano
generally employed, namely, the Broca, the Paschen and'
the Thomson galvanometers. These are however very much suscep
tible to external magnetic disturbances and can be successfully used '
only by skilled workers.
The Linear Thermopile — The extreme sensitiveness of the galva
nometers mentioned above enables us to work with only a few i
of thermoelements. The hot junctions are all arranged, in a vertical
line. The wires are very fine and are wound"
on n small frame. This is called the linear
thermopile and is used for investigating the
lines of the infrared spectrum,
22. Crookes' Radiometer, — This consists
of a number of thin vertical vanes of mien sus
pended at the ends of a light aluminium rod
r inside an evacuated glass vessel (Fig, 12).
Two such rods fixed at right angles are shown
in the figure. They are suspended in such a
can rotate about the vertical
 a. The outer face of the vanes is coated
b lampblack, while the inner faces are left
clear. When radiation (thermal or light)
falls on the blackened face the vanes begin
to revolve in such a direction that the black
ed fare continually recedes away from the
source of radiation.
The cause of this motion is easily under
stood. The blackened face absorbs the Inci
dent radiation and thereby its temperature
is raised, while the clear face remains at a
lower temperature. The molecules bombard
ing the blackened face therefore become more
hen ted than the molecules bombarding the
other side of the vane and consequently they exert greater pressure
on the vanes. The result is that there is a net effective force repelling
the vanes from the incident radiation. This is known as the radio
meter effect, and is essentially due to the presence of molecules. It is
obvious that the velocity of revolution will be a measure of the in
tensity of radiation.
By suspending the vanes by means of a quartz fibre as in galvano
meters we can measure the intensity of radiation if we observe the
. 12.— Crookes'
radiometer.
W
xi.j
THE BOLOMETER
i .j Wi a radiometer has been employed
steady deflection f^ft**;*^ ™X*#J of radiation.
pUtinuSr strips or wires when .heaKd. ,,. thc ^native
P Tire WtiveaeB rf thebolo *«e r d^penc... ^ ^ fc
ness of the galvanome er, W "° ™\ e the rise in .emperaM* and
Figs. 13 and 14,The Surface Bolometer.
(2 ) linear bolometer to neueftt of distribution of energy in the
spectrum of a black body. f evceedinsrlv thin strips of
method of constructing such a
thin conductor is as follows :— A
sheet of platinum is welded to a
thick sheet of silver and thc com
posite sheet is rolled. The sheet
is then punched out as shown in
Ficr 13, and attached to a hollow
frame of slate. The ^lve
dissolved off in nitric acid, tne
end joints being protected by a
coating o£ varnish. The strips are
then coated with platinum black.
Fitr 14, shows a front view ot
such a grid. A grid so const
ed has a resistance of about W
° hm For experimental purposes n^
fe^t e^'otte^'ntected in the form of a Wheatstone
800
RADIATION
bridge. The method of connecting the grid is shown in Fig, 15. The
grids 1 and 3 are in opposite arms of the Wheatstone bridge and are
so arranged that the strips in 3 receive the radiation passing between
trips in I and so die effect is doubled, 2 and 4 are similarly
arranged but a] cted from radiation. The whole is enclosed in
a box, In the absence of incident radiation the galvanometer shows
no deflect ion. When radiation is incident on grids 1 and 3 defection
i] oduced,
In the linear bolometer a single narrow and thin strip of plati
num is as
24. The Radiomicroraeter. — ThJ vented by Boys and is
essentially a thermocouple without an external nneter. A
single loop c of fine copper or silver wire is suspended (Fig. 16'i
between the poles of a strong permanent magnet NS as in a suspended
coil galvan To the lower ends of the copper wire two thin
bars of antimony So and bismuth Bi
are attached, and the lower ends of
these are again attached to a thin disc
d or narrow strip of blackened copper.
To the upper end of the copper lo
is attached ;i thin h g carrying
a light galvanometer mirror m few
measuring deflection, the glass rod it.
attached to a fine quartz
pension q. The whole system is ex
uely light. Radiation, falling hori
i tally on the copper disc, beats the
junction of antimony and bismuth and
an current flows through the
copper coil causing a deflection which
depends upon the intensity of the
energy re tivcd by the copper disc, To
prevent disturbance due to diamagnet
ism of bismuth, Che rods of the thermo
couple are surrounded by a mass of
soft' iron. The whole suspended sys
tem is enclosed in brass (shown shaded
in the diagram),
25. Pressure of Radiation. — As ra
diation has been shown to be identical
with light, it possess:: :il the properties
which are ascribed to light. One of
the properties of light most important for our present purpose is
it exerts a small but finite pressure on surfaces on which it is
incident. This had been suspected by philosophers for a long time
i he days of Kepler, who observed that as the comets approach
•Reproduced from "The Theory of Heat" by Preston by the kind pcrmb
> i .',',._■ <:■ j, I; .Till.i:: & Cb.
Fig. 16,* — Boys" Ri
micrometer.
PRESSURE OJ KAHM '
SOI
the sun, the tail of the comet continuously veers round, so as
to be >pposate the sun (Fig. 17). This he tried to explain
on the assumption that light exerts pressure on all material bodies
hich it is incident, but th< Lre increases in importance only
the size of the particle is reduced. The die sun
cometary matter, either dust particles or atoms, which are then
repelled by lightpressure and thus lorm die tail. Though painstak
ing experimental investigations failed to show the existence ot pressure
"it., Maxwell propounded in 1870 his Electromagnetic Theory of
Light, lowed that even on this theory Light should exert a
ire but this is very small, being equal to the intensity divided
by the velocity of light or Co the I jydensity. Calculation shows
that the pressure due to sunlight is equal to 45 ;spercm* a
Fig, 17,— Tail ot the comet'
Bartoli also showed from thermodynamic considerations that
radiation should exert some Finite pressure. The pressure of radiation
is however so small that for a long time it battled all attempts to
measure it. The difficulties were overcome only in 1900 when
Lebedew, ant! a little later, Nichols and Hull demonstrated c
mentally its existence and were able to measure it. They confirmed
the theoretical conclusion that pressure is equal to the energy density
of radiation.
When the radiation is diffuse, it can be shown that pressure is
equal to onethird the energy density of radiation.
The fact that radiation exerts a finite pressure, however small
it may be, is of great importance in the theory of blackbody radiation.
It shows tha ick radiation is just like a gas, for it exerts pressure
%\[>.
RADIATION
CHAP.
EXPLRIMENTAL VERIFICATION OK STEFAN S LAW
SOS
and possesses energy. In fact we can regard the black radiation as a
thermodynamic system and calculate its energy and entropy, and apply
the thermodynamic laws and formula:. We shall make use of these
i in the next section.
26. Total Radiation from a Black Body,— The StefanBoltzmann
L aWt — As already mentioned in sec. 7, J. Stefan, in 1 870., deduced
kallv from the experimental data of Dulong and Petit that the
total radiation from any heated body is proportional to the fourth
power of its absolute temperature. In 1884 Boltzmann gave a theore
tical proof of the law based on thermodynamical considerations. lie
red that the law applies strictly to emission from a black body.
The law is therefore generally known as the StefanBoltzmann law,
and may he formally' enunciated as follows : — If a black body at
pemture T be surrounded by another black body at
lute temperature T the amount of energy E lost per second per
of the former is
E = *{T*Tf) t .... (8)
where <r is called the Stefan's constant.
For proving this law, we consider radiation in a blackbody
chamber and apply the thermodynamical laws to the radiation as
mentioned in the previous section! Let u denote the energy density
of radiation inside the enclosure, V its volume and p the pressure of
ion. Then both U and p are simply functions of the absolute
erature T. We h ther the total energy U of radiation
equal to uV. Applying* equation (61) p. 267 we get
&t. ■ ■ ■ <9)
and
1 =
It
tdU\ A u
(10)
or
or
du _ .
tt
the radiation is diffuse (sec 25) . Hence equation (9) reduces to
T du u
3 ar "" T
,ir
T *
u = aT*, [11)
where a is a constant, independent of the properties of the body.
ce the total energy lost on one side by emission will also be pro
portional to the fourth power of the absolute temperature. That is
the StefanBoltzmann law.
27. Experimental Verification of Stefan's Law.— The law was sub
jected to experimental test by various investigators, Lummer and
♦Boltzmann deduced the law by imagining the_ radiation to perform a Camat
cycle. i :, 'd the conceptions of Bnrtoli. This is however unnecessary here
for we have shown black radiation to be analogous to a gas and cm therefore
:1 directly to apply the general thermodynatnieal laws to radiation,
Pringsheirn investigated the emission from a black body over the
range of temperatures lOCPC to 1260*0 and found the law to hold
true within experimental errors, We give below a brief account of
their apparatus and arrangement.
A is a hollow vessel containing boiling water (Fig. 18) which
acts as a standard source of radiation for calibrating the bolometer
from time to time. The black body C employed for the ran;
temperatures 2Q0°C to 600 °C consisted of a hollow copper sphere
blackened inside with platinumblack, and placed in a' bath of a
mixture of sodium and potassium nitrates which melts at 2I9°C.
This salt bath could be maintained at any desired temperature.
The temperature could be measured with a t hennaelement T.
18. — Lummer and Pringshcim's apparatus for verify trig Stefan's law.
For temperatures between 900 'C and 1300 Q C the black body
shown in Fig. 19 was employed. D is an iron cylinder coated inside
with platinumblack and en
closed in a double walled gas
furnace. The temperature
inside the iron cylinder was
obtained by a therm oe le
nient enclosed in a porcelain
tube passing through the
furnace.
The measuring instru
ment shown at B was the
surface bolometer of Lummer
and Kurlbaum. A descrip
tion of this as well as the method of connecting it has been already
given in sec. 23. Besides there are a number of watercooled shutters
so that the radiation can be stopped or allowed to fall at will.
The procedure adopted was as follows: — The hath was heated
up to the desired temperature and maintained steady, and then the
Fir;. 19.— Bl: I: , for 9£»* to 13D0 B C.
304
RADIATION
CHAP
shutter was raised to allow radiation from C to fall on the bolo
meter, and the maximum deflection registered by the galvanometer
noted. The bolometer was kept at different distances from the black
v and the inverse square law verified Next observations were
n with the black body at different temperatures. The observations
were all reduced to a common arbitrary unit depending on the radia
tion from the black body A at 100°C. and kept at a dis ■ of 63$
. Tf d represents the deflection oF the galvanometer needle,
T the absolute temperature of the black body, 290° the temperature
of the shutter protecting the bolometer, then
d =* *(r*290*) 3 .... (12)
from Stefan's law. The coefficient a was found to be constant.
Hence the truth of Stefan's law is established.
DISTRIBUTION OF ENERGY IN THE SPECTRUM
OF A BLACK BODY
28. Laws of Distribution of Energy in Blackbody Spectrum.—
From a study of the colour assumed by bodies when their D lire
is gradually raised (see p. 281) it will be obvious that as the tem
perature of a body is raised, the colour emitted by it becomes rich
in waves of shorter wavelength, Tn fact the wavelength for which
the intensity of emission is maximum shifts towards the shorter
wavelength 'side as the temperature is raised. These results were
also arrived at. by Wien in 1893 from thermodynamic considerations
n inside a hollow reflecting chamber. He showed
thai contained in the spectral region
included within the wave lengths A and A + dX emitted by a black
at temperature T is of the form
E^X=^f(XT)dX t ;i:i
and further if X m denotes the wavelength corresponding to die
maximum emission of energy and E m the maximum energy emitted.,
vr*, (i4)
and E nl T~ & =B, (15)
where B and b are constants. In other words, if the temperature oi
radiation is altered, the wavelength of maximum emission is altered
in an inverse ratio. Equation (13) is known as Wien's Displacement
law.
A further step forward in developing the theory was taken by
Planck who showed from very novel considerations (which developed
into the modern quantum theory) that the energy density of radiation
inside the enclosure is given by
V^isp^r* < I6 >
where C t = 877/* c/A s and c^— ch/k = 496 b approximately. This is
EXPERIMENTAL. STUDY OF BLACKBODY l
30!
K64
90
.known as Planch's lam. We shall 310.. consider how the spectra can
be obtained experimentally and the foregoing results verified,
29. Experimental Study of the Blackbody Spectrum. — The first
aatic study of the infra lectrum was undertaken by La
who illuminated the slit of a a is of sunlight and
produced the spectrum by a prism of rocksalt. The rays were fen
•. I lens on a bolometer which was arranged in n V me bridge
adjusted for no deflection,
he spectrum 1
be produced by a Ro
grating but on account of
the considerable Overlapp
ing of spectra o£ different m \
orclers prism 1
generally employed. Lew
of quartz or fiuorite are m
generally used. It is how
ever better to use a con ^
 for focussing
the radiation.
Wien *s displacement
was subjected to a
scries of experim tests *••■
by Paschen, Lummer and
fsheim, T< u I; c n : ; and
Kurlbaum. We give below a
1 ption of the ex
periment and of E5]
Lummer and I EngsbJ im
1 ) . They used radii
ictrically heated
carbon tube and ced
the spectrum by refraction * ; '
through a fluorspar prism.
distribution of energy
was measured by means I \
linear bolometer which was
enclosed in an airtight ci
to diminish the absorption
due to water vapour and
carbon dioxide. Corrections
were applied to convert the
prismatic spectrum into a
normal one by 1 means of the
i. dispersion curve of
fluorspar. The distribution
of energy in the spectrum tor various temper: ■ sen 62! C 'K
and lGiO^'K was obi and curves plotted (Fig, 20) . The ordinates
• powers and the abscissa: w r avelenglhs. The full
■■
:..
Si,
1  ■> S Su,
Fig. 20 — Distribution of fcjjcrgy in the
1  kbody spectrum.
306
RADIATION
CHAP.
lines denote: the curves obtained experimentally while the dotted
represent the curves calculated from a semiempirical law 
by Wiem The total radiation at a temperature is : •;• the
area • een the curve and the v: varies as T 4 .
The small patches of shaded area represent the aba
ide and water vapour. ibsorption of fluorsp
at G[t where the curves are seen to end abruptly* The wavel
of maximum emission shifts towards the ' as the temperature
rises. From these curves the values of E m and X m could be read.
The experimental data are given in Table 1.
Ta hie 1<—Experim gj ita I fe rification of Wien's
Displaceme.!
Temp. D K
X m in f/s
i'ia
A m T in
ii lie ton
e "ees
1017
1646
1.78
270.6
2928
2246
1460.4
2.04
145.0
2979
2184
1250.0
2.35
68.8
2959
2176
1094.5
34.0
2966
21 64
.
21.5
956
2166
908,
13.se
2208
7230
4.08
.
2950
2166
621,2
4.53
2.026
2814
2190
i — 2940
2188
The i ly the vi ions (14) and i
the valu ws enable us to determine the
of any sub':' umed to be a black body) if X„
, be found out. The value of 6 is seen to be .294 cm X di
\  degree.
This furni' with a simple method of determining the tem
perature of the heavenly bodies. Thus Cor the moon X^ = 14^, and
lience
T= rT . \ i lx . ■=210°K.
Similarly we can calculate the temperature of the sun (sec,
PYROME1RY
39. Gas Pyrometers. — In Chapter I we have already descri
hydro :;i thermometer having a platinum
um bulb. For temperature above B00°C hydrogen cannot be used
as the thermometric gas because the platinum bulb is permeable to
it, hence nitrogen is invariably employed. As regards the material
of the bulb Jena glass can be used to about 500*0 while quartz
can stand up to about 1300°C but is attacked by traces at alkali from
XL]
RADIATION PYROMETER
307
hand etc. Ordinary porcelain is porous and permeable to the theimo
L but glazed porcelain was formerly employed up to about
L000°C ; ' hich however the glazing softens and brea u its
expansion is not regular. Platinum alloys are now invariably used.
ln P Xv o< platinum and iridium was employed by Holborn and
the bulb brittle. Holborn and Day found the alloy of 801 t20Rh best
£r the purpose u d a modified Lorn, ol this material to
mt lfiOO'C. The} performed experiments with great rare and took
It 25  ;J L , •; vor so that their value, are
ed as standard at high temperatures. The correction however
becomes enormous at these high temperatures and hence there is
considerable uncertainty in determining the high temperature hxed
points with gas thermometer.
31. Resistance Pyrameters.The platinum resistance PF°"^er
has been described in Chapter I. It can be W& to about U
though the melting point or platinum is 1770°. It used above 1000
however, the platinum undergoes some change and does not return
to its initial zero and has to be restandardised. _ The mica insulation
; , sometimes breaks due to moisture getting inside.
32. Thermoelectric Pyrometers.— The tbermoek ■ roraeters
have been completely described in Chapter I. It is shown there
that, for temperatures up to 600°C the base coup es O ucons tanta n,
i are the most sensitive. The Pt OOPtlOUh couple i
however best for all high temperature work and can be used to about
A
mo
lOCr couple up to 1500°C when the latter is much more sensitive,
couple of Tr, QOIrlORu can be used to about 2100*C while ther
couples of tungsten and an. alloy of tungsten with molybdenum have
been used up to 3000 °C.
RADIATION PYROMETER
33. Temperature from Radiation Measurements.— In flu
fore
went we can find its temperature. We way cither measure the
total radiation emitted by the body and deduce the temperature by
making use of Stefan's law (equation 8). These are called lotal
Radiation Pyrometer*/ Or we may measure the energy i
imiti :d in a particular portion of the spectrum and make use of
Planck's law (equation 16) of distribution of energy in the spectrum.
These are called Spectral or Optical pyrometers.
Radiation pyrometers possess the great advantage that they can
be employed to measure any temperature, however high that may
be or wherever the object may be. The pyrometer itself has not to
SOS
RADIATION
[CHAP.
be raised to that temperature nor need it be placed in contact with
the hot body. Farther there is no extrapolation difficulty as the
radiation Eormube have been found to hold rigorously for ail tempera*
lures. But the radiation pyrometer suffers from a serious drawback.
It can measure accurately the temperature of ijodics only.
It is, however, generally employed to measure the temperature
of any hot source. In that case it \ at temperature at which
a perfectly black body would have the same intensity of emission
(total or spectral) as the body whose temperature is being measured.
This temperature is called the . *erature of the
substance and is conseqi lower in all cases than its actual
temperature. The greater the departure from perfect blackness the
greater is the error involved. The lower practical Limit for radiation
pyrometers is about 600°C, for then the emission from substances
becomes too small to be measured accurately. Still, however
certain devices it can be used to measure much lower temperatures
as that of the moon.
34. Total Radiation Pyroirtelcrs, — Tery was the first to devise a
radiation pyrometer based on Stefan's law. These pyrometers are
merely thermopiles so arranged that the readings are independent of
rne dflsta itween the hot body and the pyrometer. We shall
describe Very mirror pyrometer which is typical of this class. Fig. 21
shows a modern type of
ToMilUfaaioH j mh Radiation
incident from the right
side falls on the concave
or M which can be
moved backwards and
forwards for the purpose
of focussing the radia
tion on the receiver S to
which the hot junction
of the thermocouple is
attached. The cold
lion of the thermo
couple is protected from
radiation by the tov
T and is further surrounded by the box B which also contains the
thermocouple receiver, and a small opening just in front of S. The
electromotive force developed is read on a mtilivoltmeter connected
indicated in Fig. 21. The instrument possesses no lag. the steady
state being reached in about a minute. To enable the observer to
radiation pyron
ipening
being Left at the centre of the mirror to allow the incident radiation
t. pnss. Now if the image of a straight line formed by the concave
or does not lie in the plane of die inclined mirrors thev will form
two images separated by a distance (see Fig, 22) and the line will
si.
OPTICAL PYROMETERS
309
appear broken when seen through the eyepiece. The concave mirror
is moved till this relative displacement of the two halves of the linage
disappears, and then the apparatus becomes adjusted. It will be
en that so long as the heat image formed by die concave
mirror is larger than the" hole, the thermocouple measures the intensity
e heat image ot the total radiation. For if the distance
of the object is doubled, the amount of radiation failing on the mirror
r V:ln
Fi?, 22. — The focussing device.
is reduced to onefourth, but as the area of the image is also simul
Lisly reduced to onefourth the intensity is unaltered. Thus the
indications of the instrument are independent of the distance. Hence
in actual use it is essential that the object, whose temperature is to be
measured, should be sufficiently large and should be placed at not
very great distance in order thai its image is always bie an the
aperture in the box which in ordinary instruments is about 1,5 mm,
in diameter.
The E.M.F. of the couple in these cases is given by the relation
V = a(T°T l ) > .... (17)
where T is the temperature of the black body to be measured, T Q the
rature of the receiver S and b a constant which varies from 3,8
to 4,2 depending upon the instrument. Generally T can be negl
in comparison with T. This departure from the index value of 4 is
clue to various ran,:;, It is for this reason that the pyrometer has
to be calibrated by actual comparison with a standard thermometer
j radiation from a blackbody chamber or heated strip.
3S. Optical Pyrometers, — In these, the intensity of radiation
from a black body in a small width of the spectrum lying between X
and A  d\ is compared with the intensity of emission of the sine
colour from a standard lamp. The formulas required for this case can
be easily worked out by assuming Planck's law. There are two typ ,
of optical pyrometers : — (1) the Disappearing Filament type, (2) the
Polarising type. We now proceed to describe these instruments.
310
RADIATION
[CHAP,
XL]
RADIATION FROM THE SUN
311
36. The Disappearing Filament Type.— This type of pyrometer
trst introduced by Morse in America. It was later inrproi
HTi and Kurlbaum, and by Mendenhall and . A pyro
r of this type is shown in Fig, 23 and is essentially a tele
ig a lamp at the position usually occupied by the c
a metal Lube containing tbe filament of a lamp L which is heated
by the battery B and the current can be adjusted to any amount by
rying the rheostat R.
C Radiation from the source
whose temperature is re
quired is focussed by the lens
on the lamp T, where a
heat image is formed. The
lamp is viewed through the
eyepiece E in front of which
is pi need a red filter gl
"ties there are a number
of limiting rms. In
periment the
 looks through E
and varies the current in the
till the filament becomes invisible against the image of the
source. If the current is too strong the filament stands out
:;e current is too weak, th ' nt looks black. The filter
g to be done for approximately m
• '  . ■ i i I [n  O ' :
v,' nh Li pie and then extrapolated by t
rom the strength of the current required to
matc l : ! the incident radiation can be
directly calibrated in degrees.
37, Polarising Type.— In 1901 Wanner cons;. mother p)
meter in which the comparison was made by the aid o£ a polarising
Fig. tfiug Filament 'Pyro:.
L,
Fig. 24, — The Wanner Pyrometer.
device. Here the ray of a particular colour obtained from the source
is compared with a ray of the same colour obtained from a standard
electric lamp. A diagram illustrating the essential parts of the ins
trument is £iven in Fig, 24. a, b are two circular ho es arranged
optical axis of the system. Radiation b
the ; urce enters the ■■ trough a, while the comparison
:n is supplied b. which illuminates the rig
I the latter directs the light on to b, Both beams
are rendered parallel by means of an ach \c lens L 1( winch is
ced ai a distance from J t equal to its focal length, lhe
Llel beams nre dispersed by the direct vision spectroscope fc> and
tirough the polarising Rochon prism R which separates
each beam into two beams polarised in orthogonal planes. B is a
biprism placed in contact with a second achromatic lei
f n r, :  ams on the slit D 2  The biprism produces in
aos deviations of such amount that one image from each source
is brought into juxtaposition. Since the holes a, b are at the focus
of the lens L 3 the images produced by md lens are also circular
but the biprism • into semicircles. Six out of the ei
images are stopped out while the remaining two are observed through
If the two beams are of equal intensity a uniforj nainated
disc with a diametrical line is observed if the plane of polarisation of
nicolpri : i 3 with the plane oE polarisai
onent. Rota icol in either direction diminish^
: ;htness or one image and augments that of the other. If the
ants are of unequal intensity n: [s affected by i i
the nicol prism in either direction. A «a attac
to the analyser ' ■ rve the angle
Let the angle at which radiation from any source matches the
lump be <#>. Then it: can be shown that
In tan #=flf= . . • * ■ (IS)
where 7" is the absolute temperature and a 3 b are constants. If we
determine two values of $ corresponding to two values of T. we get
3 straight line from which the temperature for any value of $ can be
d. In practice, the pyrometer is calibrated by a direct comparison
ith a standard thermometer, and the disc is directly graduated in
deg;" 1
RADIATION FROM THE SUN
38. The Solar Constant.— The sun emits radiant energy conti
nuously in space of which an insignificant part reach: rth. But
even of this incoming radiation, a considerable portion is lost by
.: and scattering by the terrestrial atmosphere and is sent back
i the interstellar space/ The best reflecting constituents of
atmosphere are water, snow and cloud. The scattering is partly due
to the dust particles and partly due to the air molecules and is gener
,' small. Further the radiation is heavily absorbed by the earth's
■sphere, the total absorption varying from 20 to 40% depending
RADIATION
[CHAP.
the day and the season of the year, and the different
parts of die spectrum are absorbed to a different degree. We
:. Eor a more constant quantity which is furnished by
h solar radiations are received by one sq, cm, of
surface held at right angles to the sun's rays arid placx
the mean distance of the earth provided there were no absorptio
the atmosphere or provided the atmosphere were not present. This
is call . ant and is generally exp in calories
minute i shall now give a method 'of determining the
constant,
39, Determination of Solar Constant. — The Absolute Pyrheliometer.
— Among the early workers who attempted > neaaure the
constant were Wilson. The instrument;
which the solar radiations arc measured are called pyrheliometers.
We shall describe the materstir . . smpioyeti in the Astro
B
Fig. 25. — The wa1 iter.
physical Observatory of the Smithsonian Institution, Washington,
a crosssectional view of the apparatus.
ilacfcbody chamber for the reception of solar radiation, whii
further protected from air currents by a vestibule, not shown. This
is simply a hollow cyli need in front of AA. The chamber is
blackened inside and has its rear end of conical shape, ai
surrounded by water contained in the calorimeter DD which is 51
nisly by means of a stirring arrangement BB run by an el
motor from outside. C is a diaphragm of known aperture £<
ing the solar radiation. The incident radiation is completely absorbed
by the chamber producing a rise in temperature of the water contained
in the calorimeter. This rise is measured by the platinum resistance
thermometer F whose wire is carefully wound upon an insulating
frame round AA; At E is inserted a' mercury thermometer. The
XI.]
DKITKMLXATION Ql' SOLAR CONSTANT
SIS
calorimeter DD is carefully insulated from thermal effects occurring
■ 1 :
For calibrating the instrument, a known amount of electrical
energy is supplied to the manganin resistance wire G and the rise in
tempera t tire noted. Thu eat obtained by solar radiation can
compared with the heat generated electrically*
Another type of pyrheliometer is called the water flow ,
meter. these a steady stream of water hows past the absorption
chamber and die temperatur 'ween the incoming and
water is observed.
Instruments are called absolu
they measure the enerj v. For most purposes it is more con
ient to employ secondary pyrheliometers in which radiation is
;rbed by a blackened silver disc. In the 'compensation pyrhelio
meter of Angstrom there are two thin strips, of metal idem:. I :
every way which serve as the calorimetric body. One of these is
sosed to the sun while through the other, which .is shielded by a
screen, an electric current is passed. The strength of the current is
regulated that the temperatures of the two strips, as indicated by
thermocouples attached in opposition, is the same. The energy of
incident radiation Is then equal to the electrical energy suppli
Wing the length and breadth of the strips and their ion
incident radiation per unit area can be calcula
ith the help, of these instruments we are able to measure the
rati: reived per minute at the earth's surface and from this
w ? Iculate the solar constant The radiation received vari
with the time oF the day, depending upon the zenith distance of the
sun* If we assume the absorption to be due to an atmosphere
homogeneous composition, then applying Biot's law, we have
7 = I, (TV, [l\
where d is the thickness of the medium traversed, I (> , I the intensity
of (hi beam just at the beginning and the end of the medium. This
d strictly for a monochromatic ray and a homogeneous medium.
In t. e atmosphere there is no homogeneity in dust content or i
densiy in a vertical direction. In the actual' e: periment the intensity
of sdar radiation as received on the earth is oh liferent
elections of the sun on the same day with constant sky conditions.
rhei d varies as sec z where z h the sun's zenith distj 1 urther
ling as a first approximation* that k t is the sara . ! wave
hs we can write
S  S,, a«« ■, (20)
v., S represent the true and the observed solar constant'res
[y. and a is called the transmission coefficient and varies from
0.551 to 0.85. Then taking logarithm*
assu
lend
Then taking logarithms
log; S — IogS psecz log a.
I or •:•.:• ate work h or a i, not assume 1
•o . w 8 **? 08 ™ se Parately and from this extrapolated curve the vali
RADIATION
[CHAP,
Plotting fhe values of loo S as ordinate* and the corresponding values
of ser z as abscissa we obtain a straight line wjjose intercept on tne
ate axis gives log 5,> whence £<i is found,
. curate experiments give the mean value c \ the i snstant to he
1.937 i  per minute per sq, cm. The i int i found to
.enyear cycle of the sun.
40. Temperature of the Sun.— The sun consists of a central hot
linating in a surface called the
sion 'temperature of the sun' we generally mean the temperature of
c The temperature inside the central core is, how
milc h ! n this. We shall now describe some sin
for de ig the temperature of the photosphere of the sun,
on measurement of radiation.
41. Tamperatwe from Total Radiafiwu— According to cat
serrations by Abbot and others the so
mte per cm 2 . We can now find out what wouli ■ temjera
nrface free from an sre and havir
s \ om ■ v. which would emit the same tota
ceof the earth, This naturally
for the ture of the photosphere.
Lei the radius of the sun be r, a a ' of heat loft by
"
H — '\ir^<iT\.
u. If we Lejcrjbe
is R concentric with the sun (R being the d
ill be spri ad over
d per unit surface of the earth is
4irr 2
4'jtK 2
*T*,
=(f)v
(22)
Now r/R, the r an i ngulaT radius of the sun — 959" = 4.49 >(
fans ; cr— 5,77 y 1 fr 5 ergs sec. 1 cm." 2 degree*: S = 1957 cal.
cm. 3 min."" 1 . Hence we obtain after substituting these vahss in
that T — ain2 c }L.
42. Temperature from Wavelength of Maximum Emission.—
Wien's displacement law A M T = 0.2884 cm, X degree (eq i 14)
also be employed to determine the temperature of the
sun. Abbot's investigations show that A m z=475S A, U., wience
T = G059K. This temperature Is about 300° higher than flu tem
:• lire deduced from total radiation. Thii i gence is iasiiy
understood if we remember the fact that contrary to il u otion
made here, the sun docs not actually radiate like a blai I .iy.
43. Temperature of Stars. — The stars are so many suns, only
they are at enormous distances compared to the sun. The detemina
XL\
TEMPERATURE OF STARS
tion of their temperature is subject to the same uncertainties as in the
case of the i Eollowed are. almost identical ex
that her :otal radiation method fails as it is often no
line the diameter of stars. The temperatures are :'
intensity in their emission s] pplica
tion of equation (14), but the actual methods are far more complii
The temp u ' : altl othei ways 
from 2501)'" C (red stars} to nearly 30,000° C (for bluish white st
Books Recommended*
i, GlazebrooTc, A Did' ,' Applied Physics, Vol. 4, Arti
by Coblentz.
2. Planch, WSrmestmhlung or English translation by Mash
;). E. Griffiths, Methods of . • Temperature.
■f. Bur id Le Ch: telii , T " isurement of High 1
ure.
Abbot, The Sun.
CHAPTER XH
THERMODYNAMICS? OF THE ATMOSPHERE
1. ] ii the present chap; tiall consider the application of the
principles already developed in this book to the earth's atmosphere.
e doing so, however, we shall first state .some of the results
already known to us from the study of the atmosphere.
DU ution of temperature
2. Vertical Distribution of Temperature. — The results of aerolo
observations show that the temperature of the atmosphere
as we go up in the atmosphere. By combining
several observations made over a given region, the form of the curve
:i»2[C JI2208 Jos
20
Temperature*/*
1 UJvJuqo F ff. c :cmpevelaT«
taia =10*A
1 i — Vertical distribution of monthly mean temperatures ever A
g the variation of temperature with altitude is determined. This
curve varies slightly with the season, particularly at the lower levels.
the vertical distribution of the monthly mean temperature
over Agi a (lat. 17° TO'N, long. 78° 5'E) in the Uttar Pradesh of l
The fall of temperature due to a rise of 100 metres is usually
called the vertical gradient of temperature and the fall of temperature
Homed* rise in altitude is usually known as the lapsemlz of
temperature. The lapserate over any particular region varies with
altitude and there is also a seasonal variation.
3. Troposphere and Stratosphere. — From Fig. 1 it is evident that
Agra up to a height of 13 to 16 km. there is a rapid fall of
rature, above which., however, the rate of fall dirnxni
at a height of about 20 km. the temperature becomes almost constant.
xii,]
TROPOSPHERE AND STRATOSPHERE
317
i.€. t the lapserate vanishes. A similar discontinuity in the vertical
distribution of temperature in the atmosphere is noticed all over the
■ :j.Id, although the height at which it occurs is not the same ei
where. The outer shell of the atmosphere, in which the tempers
remains practically constant with vari i of height, is. gi'mi
ecial name ol •' foe Zone or Is
to distinguish it from the Tr> re or the Convecth which
is the lower portion in. which there is a considerable fall of tempera
ture with height. The surface o of these two region' i
the atmosphere, which plays a very important role in modern
atmospheric circulation, is railed the tropopause. TV i he of
the tropopause varies with the latitude. The Iropopau ■ is to
lower towards the ground as we proceed from the equator to the
poles, its heighJ being about M km. ii the equator and about 8 to 10
km. at the poles and is greater in summer than in winl
: causes of the diminution of temperature with height in the
troposphere are manifold. Let us try to explain here in general terms
why this temperature diminution occurs. The solar radiation 1
its pas^ge through the earth's atmosphere Is only slightly al
It, and the amount a isorbed d over such a larj
air that the latter is not ■■■ heated by i
radiation. In contrast to this, however, the energy received by
earth is concentrated and therefore heats its surface considerably.
ie heated surface in turn warms the air above it, parti) by contact
and partly by the long wavelength radiated by it and absorbi •' '•■ the
air. Now the temperature of afr at any height depends upon the 'total
energy absorbed and emitted by it. The tower a ton  :i ^ sri
temperatures emits more energy than it a and therefoi I
toco I :■ radiation. These two phenomena, the h iti of the earth
and the cooling of the layer above, so affect the density of the atmos
phere as to cause vertical convection, in consequence of which the
warm ascending air becomes cooled through adiabatic expansion ami
the descending air becomes heated by adiabatic com ires ion since
the pressure or the atmosphere decreases with elevation. In tin's v
the decrease of temperature with increase of elevation is established
and maintained throughout the region in which vertical Jon
takes place. We shall calculate in § 5 an expression for the lapserate.
Above n certain height, however, convection becomes feeble ami
the temperature of die atmosphere falls so touch that the heat radiated
by it becomes equal to the amount absorbed by It either directly from
the earth's radiation or from The p 'radiation. (This is be
cause the heat radiation received by the air from the earth reman
practically constant at al] available altitudes) . The temperature of the
layer remains practically unaltered at and above this b and
therefore the convection currents cease above this heij I he
stratosphere therefore is a result of the cessation of convection c
rents and of establishment of radiative equilibrium.
*This has already been discussed on pp. 311313.
318
[IMODYNAWICS OF THE ATMOSPHERE
!
4, Vertical Distribution of Pressure.— The theory of the
tton of pressure with increase of altitude is based on the applies
of the gas laws Ei losphere. Lei: ua assume thai
:re is in convei juilibrlum while the 
e is in isothermal or ra equilibrium. Od account of
convection currents the composition of *he air in the troposph
ctically the same at ail I or the tl ical calcula;
tion of pressure we can assume a mean gas constant for the atmos
pheric air For the stratosphere hou \\ mvection currents cannot
be assumed to exist since wa have assumed the layer to be isc
hence no fixed value can be assumed for the gas constant R/M. We
must therefore treat each constituent separately with its proper gas
constant.
Let dp be the decrease of pressure corresponding to an increase
vation dz. Then 'if p is the density of air at the point under
consideration we have, on equating the decrease of pressure to che
tit of die column dz f
dp^zgpdz. . ■ (1)
From Boyle's and Charles' law we have h — pRT/M where M U die
■ [ght of the gas. Hence (1) yields
( ; )
RT
m r
) p ' r\t * ■ ■
T to be constant, which is far from being the case as far
posphere t on integrating i result
P = pc r (•!)
This is known as irmula and gives the pressure p at the
height z in terms of the pressure /* ft at the earth's surra*
As the temperature of the air column is really not constant, in
actual practice the mean temperature, of the air column is substituted
in equation (4) , This equation can then be readily applied to cal
culate the difference in height between two stations when the baro
metric pressures at the two stations are known.
For calculating the distribution of pressure in the stratosphere
we have to apply equation (4) to each constituent scp
due to variation in composition M does not remain the same at all
is. We therefore obtain ibr the partial pressures the result
p = p t e'W IT ;p=Poe* >i '< ! * T . ■  (5)
and the total pressure is the sum of these partial pressures,
5. Con vective or Adialwitic Equilibrium.— We have already
in S 3 that the troposphere is mainly in convective equilibrium. Hence
XI!.]
WATER VAPOUR IN THE ATMOSPHERE
319
■ ion (4) which was deduced on the assumption that the ;.
phere I r . at rest ia not quite correct. The fact h really ■ :
mines the distribution of the atmosphf isation oJ
die condition tha
on being moved from cue place to another, shall take up the
sure in its new position without, any loss or gain of
. taction. Tin: law • ting he volume and pre
in the hould on such assumptioj :imate to the
atic law.
If the adiabatic law p = cp* holds for the atmosphere, we
da
dp A dp
iz
Hence from (1) we get
cyp
i* 
Integrating this v
: <*!)»,*
(6)
where p () is the de at zero level. This is the law according to
which the density shoi id i 11 off with increase in height in the tropo
sphere. Since >RT/M, we get on substitution in the relation
p = cfP the result
T =
R
or1
R(TrT)
■  ess
titu ting this in (6) we obtain
■ T is the temperature at height zero, and J llie mechanical
equi , alent of Heat. Now since y = c p je v wd JM {n p r v ) — R, the
: relation yields*
... (V)
Jr
10293'
Thus the temperature decreases proportionately to the increase of
height as we yo upward;:; in th< re. Substituting numeric ■'
■ find that the constant of the above equation is about
a. This value is about twice the experimentally observed tem
per;!; tamely 5°C. per km.
WATER VAPOUR IN THE ATMOSPHERE
6. Hygrometric State of air. — Water gets into the atrai
:i account of evaporation from surfaces of oceans, lakes,
could be (25), p. : :
I :i::;:lii n (23),
S or THE vi •
CHAP.
ered mountains, moist soil and, from various other sources.
iration d pends upon a number of factor:.;, na e
rature of the air, eloci y, the pressure and the amount of
i ' < i : litis. Increase of temperature ai
wind velocity increase aporation while increase of pres
sure and of moisture in the atn decreases it. But the
;ity of the air to I, r vapour fa limited and depends upon
temperature only, At a temperature i, air can hold only a o
: pour which is giv< i the satun :ipour
ure corresponding to the temperature t, and this anu it
• of temperature. In Table i w ..iated
vapour pressure of water at different temperatures and the values are
Table 2. — Maximum vapour pressure of water in mill'.
of ' ent temperatures.
D C
o
1
2
3
4
5
6
7
8
y
4,579
5.2M
6.101 : .
7.0^ ?.SI3 8.045
10
9.21
11.23
•
.
.
19.8
23.76
,
37.73
39.90 42. IS
■•v.'/
49.69
2
•1
12
79.60
#8.02
6
8
10
14
16
30
1 3.8
149.4
163,8
179.3
196,1
214.2
A 327..1 355.1
S 450.9
■:\
.
610.9 6^7.6 7C7.3 760.0
875.1 937.9 1004,4
plotted in Fig. 2, p. 321. If the air contains the maximum amount of
water vapour that it can hold, it is saturated ; if it contains a k
amount, it is unsaturated. In some cases, it may contain more than
the equilibrium quantity; it is then called supersaturated. The
eld in air may be expressed in gra a : c
cubic metre or in tern essure in millimetres which it exerts. This
is known as the "Absoluts Hum ic air. Humidity thus con,
information regarding quantity of water vapour in the atmos
The state of the ihere with regard to its actus ttent
is generally expressed by its Relative Humidity, which is tin
ctual quantity i pour present in a divert quantity of
air to the maximum quantity that it could hold if it were sal
'he observed temperature. Relative K
in • the hygrometric state of air Le., its moisture
itent and hold moisture ran be fully specified by
ire and relative humidity.
7. Dewpoir.. — If air containing moisture is proj cooled,
a temperature wilt be reached at which the moisture that it contains
X1L]
HYGROMETER
321
IQ 20 SG 40 50 60 10 W WlwC
Fig 2 — Vapour pressure euirve o£ water.
is sufficient to saturate it. This temperature is called the Dewpoint.
Any further cooling of the air will bring about a deposition oE moisture
on the surface of the containing vessel in the form of "dew". In
the largescale phenomena
occurring in the atmosphere the
deposition may take any one of
the different 'forms, viz., Bog,
cloud, rain, frost., hail, snow,
dew, etc:. It is easily seen that
the four quantities — tempera
ture, absolute humidity, relative
humidity and dewpoint are
interrelated and a determina
tiorj of only two of them is suffi
cient.
This can be easily done
with the help of Table 1. Thus
if the temperature and the dew
point are 30" and 20°C. res
tively, the saturated vapour
iure at 20 Q C, is 17.54 nun.
which gives the absolute humi
dity of the air, and the re] tive humidity (tt.H.) is (17,54/31.82) X
100 — 55%. Conversely., if the absolute 'humidity is known to be 9.7
mm. the dewpoint is seen from the table to be equal to 10°C, and
the R.H.= (9.7/3U2) ;< 100 = 30%.
8, Hygrometers. — The study and measurement of moisture pre
sent in the atmosphere is called Hygromeiry and the instruments
used for measuring the amount of moisture are called hygrometers
[Hys?o — moisture, meter = measurer) . From what has been said
above, it will be evident that besides temperature we need measure
any one of the three quantities — absolute humidity, relative humidity
and dewpoint. This gives rise to a variety of hygrometers which may
be broadly classified as follows:— (1) Absorption hygrometers such
as the chemical hygrometer, (2) Dewpoint hygrometers, (3) Empiri
cal hygrometers such as the wetanddry bulb hygrometer and the
hair hygrometer for which no complete theory has yet been worked
out but which are by far the most important.
9. The Cbemical Hygrometer, — Absolute humidity is directly
found out by means of the chemical hygrometer. In this instrument
a stream of air is aspirated slowly over drying tubes, and the gain
in weight of the tubes and the volume of air passed over are recorded.
Thus the mass of water vapour actually present in a given volume of
air is found out, and this is compared with the mass required 10
saturate the same volume of air at the same temperature. Tin's biter
quantity is given in tables.
The disadvantages of this medio apparatu
21
S22
THERMODYNAMICS OF THE ATMOSPHERE
CBAP.
and not easily portable, the experiment takes considerable time and
laborious corrections must be applied. Its chief use is for standardis
ing the simpler types of instruments in the laboratory.
ID. Dewpoint Hygrometers.— Hygrometers in which humidity
is found from a direct determination of the dewpoint are called Dew
point Hydrometers. Examples of this type are the Daniel!, the
Regnault'and the Dines* hygrometers. The essential principle under
lying all ol them is the same, viz., a surf ace exposed to air is steadily
cooled till moisture in the form of dew begins to deposit on it. The
temperature is again allowed to rise till the dew disappears. The
mean of the two temperatures at which the dew appears and dis
appears gives the dewpoint. These hygrometers are however rarely
used in meteorological work. They differ from one another in the
manner of cooling or in the nature of the exposed surface. We shall
therefore describe only one of them, viz., the Regnault's hygrometer.
11. Regnault's Dewpoint Hygrometer.— This consists of a glass
tube fitted with a thin polished silver thimble S (Fig, 3) containing
ether. The mouth of the tube is closed by a cork through which passes
a long tube going to the bottom of the ether, a thermometer with its
bulb dipping in the ether, and a short tube T connected on the outside
to an aspirator. When the aspira
tor is in action air is continuously
drawn through the ether producing
a cooling and the temperature of
the thermometer falls. The pto
cess is continued till moisture
deposits on the surface of the
thimble, and the corresponding
temperature is noted. In order to
help in recognizing the first ap
ance of this moisture by com
parison, a second similar tube pro
wl th a .silver thimble S but
without ether is placed beside it.
Next the aspirator is stopped, the
apparatus allowed to heat up and
the temperature when the dew
disappears is noted. The mean of
these two temperatures gives the
dewpoint. The dewpoint hygro
meter has the following disadvan
tages ;— (1) It is difficult to deter
mine the instant when dew appears, (2) the temperature of the ther
mometer does not accurately represent the temperature of the surface,
(B) the instrument should be used in still air, (4) the observer him
self, being a source of water vapour, is likely to disturb the readings.
Attempts have been made to minimise or eliminate these difficulties,
but it is not possible to get continuous records from such instruments.
Fiff. 3.— Regnault's
Dewpoint Hygrometer.
XII.]
WET AND DRY BULB HYGROMETER
S23
12. The Wet and Dry Bulb Hygrometer or P*yehromet«r.—
"Relative Humidity" can be easily measured by means of a wet and
dry bulb hygrometer. This consists of two accurate mercury thermo
meters suitably mounted on a frame. Round the bulb of one of these
is tied a piece of muslin to which is attached a w T ick extending down
into a vessel containing pure water. The evaporation from the large
surface exposed by the muslin produces a cooling and thus the wet
bulb thermometer records a lower temperature than the dry bulb
thermometer. In the steady state there is a thermal balance between
the wet bulb and the surroundings. The greater the evaporation the
greater will be the difference in temperature between the two. Now
evaporation will be greater the lesser the humidity of the air and thus
the difference in temperature between the wet bulb and the dry bulb
is a direct measure of the humidity. The rate of evaporation is, how
ever, further affected by the pressure and the wind; large pressure
tends to retard evaporation while large wind velocity accelerates it
The effect of pressure is however very small and may be neglected,
while the effect of wind is rendered constant by maintaining a constant
supply of fresh air.
A relation between the readings of the two thermometers and
certain other quantities can be easily found. If T, T' denote the
absolute temperatures of the dry and wet bulbs respectively, p the
pressure of water vapour prevailing in the air and p* the saturated
vapour pressure at T' s and 11 the barometric pressure, the rate of eva
poration will be proportional to
£p
H
and also to (T  T) ; therefore
p'p = AH(TT'),
(8)
where A is some constant depending upon the conditions of ventilation
and is determined from, a large number of experiments. In actual
practice Ilygrometric Tables have been prepared by the Meteorological
Office assuming the value of A for a fixed draught of air. From these
tables the pressure p of water vapour prevailing in the air can be
directly read if the dry bulb temperature and the difference between
tlie dry and wet bulb temperatures are known. Knowing p the rela
tive humidity can be found,
13, The Hair Hygrometer. — For ordinary purposes the relative
humidity can be roughly measured by the Hair Hygrometer. This
consists essentially of a long human hair from which all oily substance
has been extracted by soaking it in alcohol or a weak alkali solution
(NaOH or KOII) . When so treated die hair acquires the property of
absorbing moisture from the air on being exposed to it and thereby
changing in length. Experiments have shown that this change in
length is approximately proportional to the change, between certain
limits, in the relative humidity of the atmosphere. Fig, 4 shows a
THERMODYNAMICS OF THE ATMOSPHERE
CHAP.
hair hygrometer. The hair h has its one end rigidly fixed at A while
the other end passes over a cylinder and is kept taut by a weight or
spring. The cylinder carries a pointer
whidi moves over a scale of relative
humidity graduated from to 100,
The changes in length or the hair due
to changes in humidity tend to rotate
the cylinder and thereby causi a
motion of the pointer. The instru
ment must be frequejjtiy standardiz
ed by comparison with an accurate
hygrometer and then its readings are
reliable to within 5%.
Kg. 4. — Tl meter.
14. Methods of Causing Con
densation. — We shall now find out
under whaf conditions the water
vapour present in the atmosphere
be precipitated from it. This water
vapour can be condensed into liquid
water or solid ice if the actual vapour
exceeds the maximum vapour
pressure corresponding to the exist
ing temperature, This happens al
i exclusively when the air is
cooled down more or less suddenly
but in rare cases it may occur if the
jure happens to increase
■lIill: to some local h as
compression of saturated wate etc.
The cooling of air may take place by the following three
processes : —
(1) Due to radiation of heat or due to contact with cold bodies.
(2) Due to the mixing of cold and warm air masses,
(3) Due to adiabatic expansion caused by sudden decrease of
pressure.
The first process should have been the most effective in produc
ing precipitation had it been active in large masses of air. But air
even when it is moist, is a poor conductor and radiator of heat, so
that radiation and conduction of heat play a minor roTe in the pheno
menon of precipitation. The result o£ the loss of heat by radiation
or by contact, with cold bodies, such as the surface of the earth in
winter, cold walls, stones, etc, is the formation of mist, fog, dew, etc.
The second mode of condensation depends essentially on the
experimental fact that the saturated vapour pressure of water increases
much more rapidly with increase of temperature than the temperature
itself. Thus if two equal masses of air, initially saturated at tern
XII.]
ADIABATIC CHANCE OF HU.MlI> AIR
325
peratures t and ¥ respectively, are allowed to mix together, they will
acquire the mean temperature t m = (t  F)/2, while the mean vapour
pressure will be (e + e*) /2, where e, e' denote the saturated vapour
pressure of water at temperatures t and t' respectively. On account
of the above property, however, this mean vapour pressure will be
greater than E, the saturated vapour pressure at t m and therefore
the excess of water will condense. As an example take the following
illustration : — Let us have equal masses of saturated air at 4° and
82°C. When mixed up the temperature becomes 18°C. To saturate
the mass we require 15.4 gm. per cubic metre. The separate masses
contain 6.4 and 33,8 grams and the mixture ,/'  gm. per cubic
metre. Hence 20.1 — 15,4 = 4.7 gm. will separate by condensation.
If the two masses of air are not saturated before mixing, there
may be condensation in some cases. This will depend upon the
proportions of the mixture. If both the masses are very near the
point of saturation, then condensation may take place at some places,
and no condensation or even evaporation at others. This explains
the formation and disappearance of certain kind of clouds.
This third process is the most important because it is active on
a large scale and produces cloud and rain. When moist air is
allowed to expand adiabatically its temperature falls and some of
its moisture is condensed if the temperature falls below the dew
point. This is the process which generally takes place in the atmos
phere. An ascending current of moist air suffers a decrease of
pressure as it ascends ; it therefore expands almost adiabatically and
partis with some of its moisture. To calculate the cooling we have
mass of air,
IS, Adiabatic Change of Humid Air,— From the first law of
thermodynamics if dQ be the amount of heat supplied to a given
mass of air,
«*JT}***T
(9)
In case of a mass of saturated air rising upwards, the heat dQ
is added as a result of an amount dm of vapour being condensed.
Hence,
d(l= Ldm, (10)
where L is the latent heat of vaporization. Therefore
—Ldm = CpdT — '' '
MpJ
(II)
The total mass m of water vapour in the air per c.c« is given by
m = 0*623— xp,
P
THERMODYNAMICS OF TILE ATMOSPHERE
[chap
where 0.623 is the ratio of the molecular weight of water vapour to
the weighted mean of the molecular weight of the constituents of dry
air, e the vapour pressure, and p the pressure of the dry air and p
its density. Hence
dm de dp
m e p
Substituting this value of dm in (11), we have
de I.m . RT
Lm T +  dp  H dT+ j^jdp  0,
(12)
or,
Now dp=i pgdz = —jppdz* Substituting this value in (13)
we get
dT = _ s \ RT ^ Jf
dz , Lm de
e >+TdT
(14)
This is the rate of decrease of temperature with elevation of
saturated air. All the quantities on the righthand side of this
equation are known, so that — — can be easily evaluated.
Books Recommended.
1. Humphreys, Physics of Air.
2. Brunt, Meteorology.
3. Lempfert, Meteorology,
4i Hann, Lehrhuch der Meteorologie.
5. Wegener, Thermodynamik der Atmosphere.
APPENDIX 1
ERRORS OF MERCURY THERMOMETER AND
THEIR CORRECTION
As mentioned on p. 2 various corrections must be applied to
the mercury thermometer if it is used for accurate work. The method
of applying these corrections is explained below : —
(i) Secular Rise of Zero. Glass is to some extent plastic and
therefore its recovery to its original volume is an extremely slow pro
During the construction of the thermometer the glass is heated
to high temperatures and then allowed to cool. In this cooling pro
cess the contraction of the glass first takes place rapidly and then
slowly even upto several years. Naturally therefore when calibration
of the thermometer is usually undertaken the glass has not contracted
to its final steady volume and the zeropoint shows a secular rise for
years due to this gradual contraction. This defect can be greatly
removed by choosing suitable material for the glass of the thermo
meter, by properly annealing the tubes and storing them for years
before making thermometers out of them.
(it) Depression of Zero. This defect is also due to the defect in
the property of glass mentioned above. When a thermometer is
suddenly cooled from 100° to 0°C, the bulb does not at once regain
its original volume and there is a consequent depression of zero, 
magnitude is greater the higher the temperature to which the ther
mometer was exposed and the longer the duration of this exposure.
The method adopted by the Bureau International to correct for the
depression of zero is the "movablezero method" of reading temper a
tures. In this method the boiling point (100°C) is first determined
and immediately after, the ice reading is taken ; let these readings on
the thermometer be X and Z respectively. Suppose this thermometer
reads X s when immersed in a bath at (°C. Immediately after this
the thermometer is immersed in ice ; let its corresponding reading
be Z ( , Then the correct temperature f°C of the bath is given by
xZ
100,
(Hi ) Errors in the fixed points. For the lower fixed point the
thermometer is clamped vertically with the bulb and a little part
of the stem surrounded by pure ice mixed with a little quantity of
distilled water. Suppose in the steady state the mercury stand's at
0.PG ; then the freezing point correction Is j0J o C (additive). If
the level of mercury stands above the zero degree mark, the correc
tion is subtractive.
For the upper fixed point the thermometer is kept suspended
inside a hypsometer with the bulb exposed to steam in the inner
chamber. The steady reading of the mercury level is observed and
the reading of the manometer indicating the pressure of steam noted.
Suppose the thermometer reads no .2°C"under a pressure of 75.8 cm
328
APPENDIX
of mercury. Since the boiling point for this pressure is 99.93°C, the
correction is 99.93 99.2 = 0.73°G and is positive. If the obse
boiling point i.s above the calculated one, the correction is negative.
(jV) Correction fornonitnift ire. As capillary tubes are
drawn and not bored, slight inequalities in the diameter of the bore
are bound to exist in the atem of the thermometer. This necessi atea
a small correction which is carried out as follows : — A small portion
of the mercury thread is detached from the rest and its Le
measured when it occupies successively different parts of the stem,
say between and 10, 10 and 20, ♦ . , 90 and 100 marks. The
measured lengths will vary from place to place due to nonuniformity
of the bore ; let these lengths be t it l 2 lw> respectively. Let the
corrections to be made for nonuniformity in the vicinity of the
0, 10, 100 mark be a^, «ki «iocj respectively. If I is the accurate
length of the mercury thread used,
J = / x t a 10  % . . . . (1)
J = 'a + *ao»io ■ ■ • • ( 2 )
I = JiO + «1W»  «B0 ■ • • (3)
Adding up,
10 I = & f J s + r 10 ) H C^joo  flo) • • • (4)
a 1OT and a n are the corrections to the upper and lower fixed points
which can Trained experimental!) as explained in (Hi) above.
Hence / can be calculated from ('!) . Substituting this value of / in (1)
«,o can be calculated since a (t is known. Similarly from (2) a
then di awn with the marked
divisions i . a vil ..... a vl[ as ordinate. From this
.i Lures can be easily rend,
In li thermometers this tedious correction has
not to be applied by the user as the interval between the fixed points
is subdivided not into equal parts but into equal volumes to represent
the degrees on tins thermometer.
(v) Correction for lag of the thermometer. If the bulb of a
thermometer is placed in a hot bath, the thermometer will not attain
the temperature of the bath instantaneously will require a small,
definite interval of time to attain that temperature. This is called
the "lag" i I the thermometer. The lag of the mercury thermometer
increases with the mass of mercury and the thickness of the glass
and also depends upon the nature of the medium surrounding the
bulb. Due to this lag the thermometer reading will be higher when
a bath is cooling and lower when the bath temperature is rising.
Suppose we consider the case when the temperature of the bath
is rising. The correction for lag is applied as follows: — The bulb of
the thermometer is immersed in the bath and the thermometer read
ings noted as a function of time and the observations plotted on a
graph with temperature as ordinate and time as abscissa (Fig. 1) .
The curve AB represents the rise of temperature with time. To
ERRORS OF MKKCLRY THERMOMETER
: 'J
find the correct temperature of the bath at each point on the curve,
an auxiliary experiment is performed in which the bulb of the thermo
meter is immersed in a thermostatic bath maintained at a temperature
somewhat higher than the maximum recorded in the main experiment.
The thermometer readings are read at short intervals until the thermo
Tlrne
A" ^
Time
Fig. 1
Fig. 2
meter attains the steady temperature of the bath. These readings are
plotted on a graph with time as abscissa and the difference of the
h; Miometer reading and the temperature of the bath as ordinate,
and the curve RS (Fig, 2) obtained.
I l will now be assumed that the lag does not depend upon the
actual temperature of the bath but only upon the rate at which
the temperature is changing. Consider a point M on the main
curve. The lag at this instant depends upon the slope (dO/dt) of the
curve at this point M. Now find out a point P on the curve RS where
the slope is the nine as the slope at the point M, Then the lag PQ
will also be the lag at M. Hence if PQ s= MN, N gives the correct
temperature corresponding to M. In this way the corrected curve
A'B* can be easily drawn.
(vi) Error due to changes in the size of the bulb caused by
variable internal and external pressure. For diminishing the time
lag the bulb of the thermometer is usually made thin, Am increase
in external pressure therefore easily alters the volume of the bulb
and causes a rise of mercury level in the stem. Suppose the thermo
raduated when the external pressure is equal to the atmos
pheric pressure, If the external pressure is now* increased the bulb
will contract and the mercury En the stem will rise. The external
pressure coefficient is defined as the ratio of the rise of mercury in the
expressed in degrees, to the increase in external pressure,
expressed in mm. of mercury. This ratio can be easily determined
experimentally. Knowing this and the external pressure to which the
bulb is subjected at the time of reading the thermometer, the correc
tion to be applied can be readily calculated.
When a thermometer has been graduated in the horizontal posi
tion and a subsequent reading is taken with the thermometer in the
vertical position, there is an increase in the internal pressure due to
the vertical column of mercury in the thermometer. The bulb con
330
APPENDIX
sequently expands causing a depression of mercury in the stem, for
which a correction is necessary. To determine this correction the
readings of the thermometer are observed in the horizontal and vertical
positions at any one temperature. The difference in the two readings
gives the depression of mercury in the stem due to an increase in
pressure caused by the mercury column which ex Lends from the centre
of the bulb to the mark on the thermometer at which mercury stood
in the vertical position of the thermometer. Expressing this pressure
in mm. of mercury we can define the internal pressure coefficient as
the ratio of the depression of mercury in the stem, expressed in
•rces, to this increase of internal pressure. Knowing this coeffi
cient, which is constant, the correction can be calculated for any read
ing of the thermometer in the vertical position.
(vii) Error due to capillarity. The surface tension of mercury
causes an excess of pressure within trie meniscus over that outside.
This excess pressure depends on the radius of the tube at the point
where the meniscus lies {p cc 1/r). If the stem is not uniform in
bore, there will be variations of internal pressure as the thread of
mercury rises or falls and therefore the thermometer readings will not
be very accurate. Further the angle of contact beween mercury and
the sides of the tube depends upon whether the mercury is rising or
falling, the meniscus being flatter when mercury is falling. Therefore
a rising thread always gives somewhat lower readings than a falling
one. it is also found that the mercury thread is less disturbed by
capillarity when it rises than when it falls and therefore it. is preferable
to take readings with a rising column.
(viii) Error due to exposed stem or emergent column. Generally
when the temperature, of a bath is measured, only the bulb of the
thermometer and a portion of the stem are immersed in the bath.
Tn such rnses the part of the stem exposed to the atmosphere does
not acquire the temperature of the bath and therefore the thermometer
reading will be less than the true temperature of the bath. The
correction for this exposed or emergent column is applied as follows: —
Let the thermometer reading he \ when the stem upto t 2 mark is
immersed in the bulb. Thus n (■= f a — 1 2 ) divisions are exposed to
the atmosphere, and its average temperature to is measured by a
special integrating thermometer with a long bulb placed near it, with
its centre coinciding with the centre of the exposed part. Let t
denote the corrected temperature and B = m — g the coefficient of
apparent expansion of mercury in glass (m = expansion coefficient of
mercury, g, of glass) . Then a mercury column w T hose length is n
divisions at * has to be corrected to the temperature t. If this
column were to rise in temperature from t to *, the increase of height
(which measures the increase of volume) would be «*{*—/<>) divisions.
This must therefore be added to the observed reading t t to give t,
Hence
QT
f a — n8t
1—nB
EXAMPLES*
I
1. Discuss the advantages of using one of the permanent g
as a thermometric substance for defining a scale of temperature.
Describe some convenient and accurate form of gas thermometer,
explain its mode of use and show how the temperature is calculated
from the observations made with it. (Madras, B.Sc.)
2. The pressure of the air in a constant volume gas thermometer
is 80.0 cm. and 109.4 cm. at 0° and 1()0°C, respectively. When the
bulb is placed in some hot water the pressure is 94.7 cm. Calculate
the temperature of the hot water,
5, An air bubble rises from the bottom of a pond, where the
temperature is 7°C,, to the surface 27 metres above, at which the
temperature is 17°C Find the relative diameters of the bubble in
the two positions, assuming that the pressure at the pond surface is
equal to that of a column of mercury of density 13.6 gm. per cc. and
76 cm. in height,
4. Explain clearly the meaning of absolute temperature of the
air thermometer scale. A gram of air is heated from 25 °C to 70°C.
under a constant pressure of 75 cm. of mercury. Calculate the
external work done in the expansion given that the density of air
at N.T.P. is 0001293.
5. What is an air thermometer? Explain the method ol
measuring temperature by Callendar's compensated air thermometer,
Describe a method for measuring very high temperatures. (A. IT.,
B.Sc.)
0. Describe the Callendar's compensated thermometer and
explain how temperatures are taken with it. (A. LL. B.Sc, 1944 :
Utkal Univ., 1952; Punjab Univ., 1954.)
7. Describe briefly the method of standardisation and the range
of usefulness of platinum resistance thermometers, and discuss some
of the difficulties of precise resistance measurement and the precau
tions to be taken to avoid or correct for these, (A. U„ B.Sc. lions,,
1929.)
8. Give an account of the construction and use of the platinum
resistance thermometer, pointing out any special advantages of the
instrument. (Utkal Univ., 1950; Gujerat Univ.. 1951 ; Punjab Univ.,
1956.)
9. Describe various methods of measuring high temperatures.
(A. U., B.Sc, 1931, 1932, 1949; Gujerat Univ., '1951; Punjab Univ.,
1956.)
10. State, with reasons, the type of temperature measuring
device which you consider most suitable for use at temperatures
of (a) — 20()°C. :
(London, B.Sc.)
(b) — 50*G„ (c) 50°C., (d) 700°C> (e) 200 a C.
♦These examples have been classified and arranged chapterwJse correspond
ing to the twelve chapters of the book.
332
EXAMPLES
11. Describe two methods for measuring high temperatures.
State: clearly the principles underlying them and the range and sensiti
vity of each. (A, V» B. St, 1941.)
12. Write a short essay on the measurement of (a) high and
(6) low temperatures. (Delhi Univ., 1954; Punjab Univ., 1946.)
iS. In what respects is a constant volume gas thermometer
superior to the constant pressure gas thermometer and the mercury
inglass thermometer. Describe the construction and use ot the inter
national standard hydrogen constant volume thermometer. (Patna
Univ., 1949; Utkal Univ., 1954,)
II
1. Enunciate Newton's law of cooling and show how corrections
can be made for the heat lost by radiation during calorimetric experi
ments. Establish a relation for finding the specific heat, of liquids by
the method of cooling. (Nagpur, B.'Sc.)
2. If a body takes B minutes to cool from 100°C. to 60 C C„ how
long will it take to cool from 60°C> to 20*C, assuming that the
temperature of the surroundings is 10°C, and that Newton's law
•of cooling is obeyed.
3. Describe Joly's differential steam calorimeter and explain
how it. is used for finding the specific heat of a gas at constant volume.
State the corrections to be made. (Punjab Univ., 1952, 1954 ;
jerat Univ., T 95 1 ; Patna Univ., 1947; Calcutta Univ., 1947.)
4. Describe the steam calorimeter. Explain how it may be
letermine (i) the specific heat of a gas at constant volume,
rific heat of a small solid. (A. U., 13. Sc.)
5. Describe a method of determining the specific beat of a gas
at constant volume, giving a neat diagram of the arrangement of the
apparatus necessary. Why is the specific heat at constant pressure
ter than that at constant volume ? (Dacca, B.Sc.)
6. Describe the constant flow method of Callendar and Barnes
for the measurement of the mechanical equivalent of heat. In an
experiment using this method, when the rate of flow of water was
i, per minute, the heating current 2 amperes and the difference
of potential between the ends of the heating wire 1 volt, the rise
of temperature of the water was 25°C, On increasing the rate of
flow to 25.4 gm, per minute, the heating current 3 amperes and
the potential difference between the ends of the heating wire to 151
volts, the rise of temperature of the water was still 2.5 °G. Deduce
the value of the mechanical equivalent of heat. (A. U., B.Sc.)
7. Describe Nernst vacuum calorimeter and indicate briefly how
it has been used for measuring specific heats at low temperatures.
(A. U., M.Sc, 19,25,)
8. Give an account of the continuous flow method of measuring
the specific heat of a gas at constant pressure and point out its
advantages. (A. U., B,Sc, 1938.)
9. Describe an accurate method of measuring the specific heat
of a gas at constant pressure. (Punjab Univ., 1941; Gujerat Univ.,
1949, 1951.)
EXAMPLES
333
'';
10. In a determination of the specific heat at constant pressure
r Regnault's method the gas is supplied from a reservoir whose
volume is 30 litres at 10°C. The pressure of the gas in the beginning
is 6 atmos. and in the end 2 atmos., the temperature remaining
constant at 10°C. The gas was heated to I50°'C. and led into a
calorimeter at 100°C. The final temperature of the calorimeter and
contents was 315°C. and its water equivalent was 210 gm. If the
ly of the gas is 0*089 gm. per litre at N. T. P., calculate its.
c heat at constant pressure.
11. A quantity of air at normal temperature is compressed
slowly, (b) suddenly, to j\, of its volume. Find the rise of
(a
temperature if any; in each case
[Ratio of the two specific heats of air = 1*4, log 275 == 4362 ;
log 6858 — 0'83fi2.]
Deduce the formula used for (b) . (A. U, B.Sc.)
12. In a Wilson apparatus for photographing the tracks of
particles the temperature 
of the air is 20°C. li its volume is in
creased in the ratio 1375:1 bv the expansion, assumed adiabatic,
calculate the final temperature of the air. (The ratio of specific heats
of air = 14L)
15. Distinguish between adiabatic and isothermal changes and
show that for an adiabatic change in a perfect gas jbu *= constant
where y is the ratio of specific heat at constant pressure and constant
volume respectively. (Allahabad Univ., 1952; Punjab Univ., 1954,
9 no.)
14. Deduce from first principles the adiabatic equation of a
perfect gas.
A motor car tyre is pumped up to a pressure of two atmospheres
at 15°C, when it 'suddenly bursts. Calculate the resulting drop ir
the temperature of air. (A. U., B.Sc, 1938.)
15. Describe a method of determining the ratio of the specific
heats of a gas at constant pressure and constant volume. How has
the mechanical equivalent of heat been calculated from a k
of this ratio? (A. U, B.Sc.)
16. Explain why the specific heat of a gas at constant pressure
is greater than that at constant volume. Obtain an expression for the
difference between the values for a perfect gas. Find the numerical
value of their ratio for monatomic gases. (Punjab Univ., 1949, 1951;
Utkal Univ., 1952.)
17. Describe the Clement and Dcsonncs' method of finding the
ratio of specific heats of air, giving the simple theory of the method.
What are the objections to the method and whnt modifications and
improvements have been proposed. (A, U., 1950; Bihar Univ., 1954;
Utkal Univ., 1955; Punjab Univ., 1953, 1958.)
18. Derive the relation between volume and temperature of a
mass of perfect gas undergoing adiabatic compression.
A quantity of dry air at 15°C is adiabatically compressed to Jth
of its volume/ Calculate the final temperature given y — 14 and'
4*'* = 174 (Aligarh Univ., 1948; Punjab Univ., 1957.)
334
F.XAMl'U'5
19. Explain how the mechanical equivalent of heat can be
deduced from a knowledge of die specific heats of air at constant
pressure and constant volume. State clearly any assumptions made
in your reasoning and describe experiments, if' any, which afford
justification for such assumption. (Madras, B.Sc.)
20. Describe a method of determining the ratio of the two speci
fic heats of a gas. Show that it follows from the kinetic theory of
gases that the ratio of the two specific heats in the case of a mona
tomic gas is 1.66, (Madras, B.Sc.)
21. Find the ratio of the specific heats of a gas from the follow
ing data : — A flask of 10 litres capacity weighs, when exhausted,
160 gm. ; filled with the gas at a pressure of 75 cm, of mercury it
weighs 168 gm. The column of the gas, which when confined in a
tube closed at one end and maintained at the same temperature as
the gas in the flask, responds best to a fork of 223.6 vibrations per
second, is 50 cm.
22. Determine the ratio of the specific heats of air from the
following data :— Velocity of sound =34215 cm. per sec. in air at
750 mm, and 17°C ; density of air — ,00129 gm. per c.c, at N. T. P, ;
coefficient of expansion of air— ^ £=981 cm. /sec. 2 j density of
mercury e= 13.6 gm, per c.c, (Manchester, B.Sc.)
Ill
I.
In an experiment with Joule's original apparatus the mass of
the ; on either side was 20 kilograms and each fell through a
one metre forty times in succession. The water equivalent
calorimeter and its contents was 6 kilograms and the rise in
temperature during the experiment was 0.62°C. Calculate the value
of the mechanical equivalent of heat,
2. Determine Lhe heat produced in stopping by friction a fly
"' kilograms in mass and 50 cm. in radius, rotating at the rate
ond assuming the flywheel to be a disc mounted
axial b and having a uniform distribution of mass,
3. A canon ball of 100 kilograms mass is projected with a velocity
of 400 metres per second, Calculate the amount of heat which would
be produced if the ball were suddenly stopped.
4. In one hour a petrol engine consumes 5 kilograms of petrol
whose calorific value is 10,000 cals. per gram. Assuming that h per
cent of the total heat escapes with the exhaust gases and that 12 per
cent of the heat is converted into mechanical energy, find the average
horsepower developed by the engine and die initial rate of rise of
temperature of the engine per minute. Radiation losses mav be
ignored and the water equivalent of the whole engine is 40 kilograms.
5. Give the outlines of the methods bv which the mechanical
equivalent of heat can be determined. Assuming that for
mi no
air at
constant pressure the coefficient of expansion is 1/273, the density
at C. and atmospheric pressure is 0,001293, the specific heat.
fc
EXAMPLES
335
^==0.2389 and the ratio c p fe s = 1.405, calculate the mechanical
equivalent o£ heat. Suppose that there is inappreciable cohesion
between the molecules. (Bombay, B.Sc.)
6. Define the mechanical equivalent of heat. If the kinetic
energy contained in an iron bail, having fallen from rest through
21 metres, is sufficient to raise its temperature dirough 0.5 °C V
calculate a value for the mechanical equivalent of heat (given g = 980
cm. per sec. per sec. and specific heat of iron = 0J) .
7. Show how the method of electrical heating has been adopted
in the determination of the mechanical equivalent of heat. One
gram of water at 100°C. is converted into saturated vapour at the
same temperature. Calculate the heat equivalent of the external work
done during the change. Density of water at 100°C. = 0.958 gms.
per c.c. ; density of saturated steam at 100°C. = 0.000598 gms. per c.c.
8. Describe a laboratory method of determining J. Give expe
rimental details. (Delhi Univ., 1951.)
9. Describe with relevant theory Rowland's method of finding
the mechanical equivalent of heat. Point out the significance of the
result. (Nagpur Univ., 1956.)
10. The height of the Niagara Falls is 50 metres. Calculate
the difference in temperature of the water at the top and bottom
of the Fall if J — 4.2 X 10 T ergs/cal.
11. A lead bullet at temperature of 47.6°C strikes against an
obstacle. If the heat produced by the sudden stoppage is sufficient
to melt the bullet, with what velocity the bullet srikes the obstacle ?
It is assumed that all the heat goes to the bull el. Melting point of
lead = 327°C, specific heat of lead = .03 i ./degree, latent heat
of fusion of lead = 6 cal./gm. f J = 4.2 X 10 T ergs/cal. (U ileal Univ.,
1 950,)
12.^ Describe Callendar and Barnes' continuous flow method of
measuring the mechanical equivalent of heat. State how the method
can be adopted to measure the variation of specific heat of water
between 10 Q C and 90 C 'C. (Punjab Univ., 1951 ; A. U., 1951.)
i:>. Explain what is meant by the "velocity of mean square" of
molecules of a gas and their "mean free path". Show how thsse
two quantities can be found. (Bombay, B.Sc.)
14. Calculate the molecular velocity (square root of the mean
square velocity) in die case of a gas whose density is 1.4 gm. per litre
at a pressure of 76 cm, of mercury. Density of mercury — 18.6,
g=:981 cm. per sec, per sec. (Manchester, B.Sc.)
15. Show that pressure of a gas is equal to twothirds of the
kinetic energy of translation per unit volume. Calculate the kinetic
energy of hydrogen per grammolecule at 0°C. (A. U., B.Sc, 1949) .
of gases
this theory ? (A. U., B.Sc.
Deduce Boyle's and Avogadro's laws from the kinetic theorv
. What interpretation of temperature is given according to
17. Outline the essential features of the kinetic theory of gases
*md an expression for the pressure of a gas on the basis of kinetic
theory. (Punjab Univ., 1954, 1955, 1957; Delhi Univ., 1954.)
336
EXAMPLES
18. Show that the pressure exerted by a perfect gas is  of the
kinetic energy of the molecules in a unit volume. Explain on the
basis of the 'kinetic theory (i) why the temperature of a gas rises
when it is compressed, and (ii) why the temperature of an evaporat
ing liquid is lower than its surroundings, (Punjab Univ. r 1956.)
19. Deduce an expression for the conductivity of a gas from the
kinetic theory. How would you actually proceed to determine the
conductivity of any particular' gas ? (A. U„ B.Sc, lions,, 1931.)
20. State the law of equipartition of energy. Prove that for a
monatomic gas, the value of gamma, the ratio between the specific
heats is 5/3 and for a diatomic gas it is 7/5. (A, IL, B.Sc., 1932,)
21. l r ind an approximate expression for the mean free path of
a molecule in a gas, and give a short account of any one phenomenon
depending on the length of the mean free path. (London, B.Sc.,
Horn.)
22. On the basis of the kinetic theory deduce an expression
for the viscosity of a gas in terms of the mean free path of its mo
cules. Show that it is independent of pressure but depends upon
the temperature of the gas. (Baroda Univ., 1954,) ^
23. What is meant by (a) the "coefficient of viscosity" of a gas,
he ri'. free path of its molecules"? Show how to deduce
a relation between these quantities from the kinetic theory, (Lond.,
B.Sc.')
24. Describe some phenomena which have led to the conclusion,
that molecules have a finite diameter and mean free path. How can
atter be determined?
IV
1. How has van der Waals modified the isothermal equation for
a gas ? Calculate the values of the critical pressure, volume and tem
rature in terms of the constants of his equation.. How do the
theoretically derived results tally with experiments? (A. U„ BSc,
193.1.,)
2. Derive van der Waals" equation of state and obtain expres
sions for the critical temperature, pressure and volume in terms of
the constants of van der Waals' equation. (Punjab Univ., 1957 ;
Delhi Univ., 1953 ; Bombay Univ., 1953 ; Patna Univ., 1948 ; Calcutta
Univ., 1948 : Aligarh Univ.. 1950 ; Nagpur Univ., 1953 ; Baroda
Univ., 1954.)
3. Express the value of the critical temperature in terms of a,
b and R, Calculate its value for CO... where a — .00874 and h = .0023.
(Punjab Univ., 1957 ; Allahabad Univ., 1952.)
4. Define critical temperature, pressure and volume of a vapour
and give some account of the behaviour of a substance near the criti
cal point. (A. U., B.Sc.)
5. Draw a diagram showing the general form of the isothermal
including both liquid and vapour state, and explain the meaning of
the different parts of the curve. What is the true form of the straight
portion of this curve and why ? (A, U., B.Sc, 1935.)
rXAMPLES
m
6. Give an account of the properties of fluids in the neighbour
hood of the critical point. Describe bow you would determine the
critical constants of a substance, (A. IL, B.Sc., Hons,, 1928).
7. What is meant by the critical point in the state of a fluid ?
Show on a diagram the character of typical isothermals of a fluid
above and below the critical temperature. Explain how van der
Waals' equation accounts for the existence of a critical point.
(London, B.Sc.)
8. Explain how van der Waals' equation accounts for the exist
ence of a critical point. Calculate the values of tile critical pressure,
critical volume and critical temperature for a gas obeying van der
Waals' equation. (A, U., B.Sc, 1947.)
9. Describe the experiments of Andrews on carbon dioxide.
State and discuss the results obtained by him. Hence show that
liquid and gaseous states are only distant stages oE a long series of
continuous changes. (Bombay Univ., 1953 : Baroda Univ., 1954 ;
Punjab Univ., 19
10. Explain in brief outline the reasons which led van der
Waals to his equation {p + afv*) (v — b) =RT. Discuss how far
\i equation is in keeping with' experimental facts. (Allahabad
Univ., 1950.)
IL Describe Arnagat's experiments on die compressibility of
gases at high and low pressures. State his important conclusion*.
Indicate how the results can be explained with the help of the van
der Waals' equation. (Madras Univ.)
12. Derive the reduced equation of a gas starting from van der
Waals' equation of state. Show that if two gases have the same
reduced pressure and volume, they also have the same reduced tem
perature, (Punjab Univ.)
V
1. How would you determine the vapour pressure of a liquid
above its normal boiling point ? Explain how clouds are formed by
the mixing of warm moist air with cold moist air, (A. U., B.Sc)
2. Describe a method of determining the vapour density of a
volatile liquid, and explain the theory of your method. (Dacca, B.Sc.)
Give an account of a method which has been adopted for
the determination of the pressure of saturated vapour between I00°C,.
and I2(PC, Explain clearly what is meant by the statement that
the specific heat of saturated vapour at 100°C is negative. /Madras
B.Sc.) v
4. An electric current of 0.75 ampere is passed for 30 minutes
through a coil of wire of 12.4 ohms resistance immersed in benzene
maintained at its boiling point, and 29.85 gm. of benzene are found
to have vaporised. Calculate the latent heat of vaporisation of
benzene.
5. What are the principal differences between saturated and
unsaturated vapours? How would you determine the pressure of
saturated water vapour at temperatures between 40 °C and 11.0°C?
338
EXAMPLES
859
6, A mixture of a gas and a saturnLcd vapour is contained in
a closed space. How will the pressure of tJie_ mixture vary
(a) when the Leinperature is changed and Lhe volume is kept constant,
hen the volume is changed and the temperature is kept constant ?
7, Describe and discuss a method by which you could determine
the latent heat ol a metal which melts at about 20Q e C
8, Define latent heat of evaporation ol liquids and describe .
how it can be measured accurately. Stale Trouton's law connecting
atenl Nat of evaporation with the absolute temperature of the
boiling point.
VI
1. Write an essay on 'artificial production of cold'. (A. U.,
B.Sc, Hons., J 030.)
2. Describe the manufacture of liquid air. (A. U., Ji.Sr,, 1030.)
3. Discuss theoretically the production of cold by expansioi
gases through porous plugs. I low has iiuriple bee;
in m:: or liquefying air? (A. U., B.Sc, 1937; Dell
1953, 1954, 1959.)
4. Write a short essay on the liquefaction of the socalled per
manent gases. (Dacca, B.Sc.)
5. ig hydrogen, and explain the
principle involved in the process. (A. u., B.Sc.)
6. Describe and discuss the porous plug experiments of Joule
and Kelvin. Explain what is meant
ice to hydrogen and h<
ii the meth; in the manufacture
of I '.Sr., 193
8. Dii tiabatic change and Toule
Thomsoii cribe hov. ier has been utilised for I
\. V., B. 7.)
!i. •.: porous plug experiment of Joule and Thomson
and discuss the result ed tor differed h special refer
ence to hydrogen. Indicate hov.' the results have been utilised for
the Hi ►! ail and h] (Bihar Univ., 1951; Punjab
Univ., 19!
10. What Is JouleThomson effect? Obtain the expression for
iced assuming that the gas obeys van de
equation. Wlr, do hydrogen and helium show a heating
ordinary temperature? (Allahabad Univ., 1955; Rajasthan Univ.,
I960.) '
VII
1. Describe and explain a method of measuring the linear
expansion of solids by means of interference bands.
2. If a crystal has a coefficient of expansion 13 X 1G"" T in one
direction and of 231 X 10 T in every direction at right angles to the
first, calculate its coefficient of cubical expansion. (London, B.Sc.)
3. A lump of quartz which has been fused is suspended from a
quartz fibre and allowed to oscillate under the influence of the torsion
of the fibre. If the coefficient of linear expansion of the material Is
7 X 10 T and the temperature coefficient of its rigidity is f 13 X 1®~S
how many seconds a day or what fraction of a second a day would a
nge of temperature of 1°C1 make? (London, B.Sc.)
•1. A seconds pendulum is one which completes half an oscilla
tion in I second. Such a pendulum of invar is given and is correct
at I0*C. If the average temperature for the three months of June,
July and August is 25°C. and the clock is correct at 12.0 a.m. on
June ]st, how much will it be incorrect at 12.0 a.m. on September
1st ? Coefficient of expansion of invar is 1 X 1Q  ***
5. Describe, in full detail, the method by which the expansion
of crystals, when heated, may be studied experimentally. (Allahabad
Univ., 1943; Aligarh Univ.. 1949; Punjab Univ., 1946, 1954.)
6. Describe a method by which the cubical expansion of a liquid
can be accurately determined by weighing a solid of known expansion
in it at two known tempi
A solid is found to weigh 29.9 gms. in a liquid of specific gravity
' T:. its weight in air bei gms. It we3
in the same liquid at 25°C pecific gravity is 1.17. Calculate
the coefficient of cubical ol the
7. («) Describe K method for the determination oi
coefficient of absolute expansion of ,. Indicate briefly the pre
■ led by Regnault to avoid errors.
(h) If the coefficient ol cubical expansion of glass and mercury
1 and 1,8 what fraction of the whole
volume of a Id be filled with mercury in order that
the empty p a it should re i nsta n t when gl ass and
ed to the same i mre. (Dacca, B.Sc.)
S. Describe and explain the interference method For finding
the coefficient : expansion of crystals. Ii 1 ss the
rial expansion of a crystal differ from that of an c metal,
(Punjab Univ., 1961.)
9. Describe I' s method of finding the absolute co
efficient of expansion of nn te calculations 3
in it. Why is this method better than others? (Calcutta Univ.,
1949.)
1 ii The difference between the fixed points of a mercury therm o
is 18 cm. If the volume of the bulb and capillary tube up to
the 0° mark is 0.0 c.c, calculate the sectional area of the capillary.
■•Jem of cubical expansion of mercury == T.S X MH*> coefficient
of lii mansion of glass — 7 X JO 6 . (Madras Ur
11. An air bubble rises from the bottom of a pond, where
the temperature is 7°C, to the surface 7 metres above, at which the
temperature is 17°C. Find the relative diameters of the bubble in
the two positions., assuming that the pressure at the pond surface is
equal to that of a column of mercury of density 13.6 gm. per cc.
and 76 cm. in height.
340
EXAMPLES
VIII
1. By what processes does hot water in an open vessel lose
heat ? Describe experiments by which the several causes of loss may
be shown to exist. (Dacca, B.Sc.)
2. Define the thermal conductivity of a substance and describe
some way of finding it.
An iron boiler 5/8 inch in thickness exposes 60 square feet of
surface to furnace and 600 lbs. of steam at atmospheric pressure are
produced per hour. The thermal conductivity of iron in inchlb.scc.
Slits is 0.0012 and the latent heat of steam is 536. Find the tem
perature of the underside of the heating surface. Explain why this
is not the temperature of the furnace. (Dacca, B.Sc.)
3. Explain the difference between the thermal conductivity and
the diffusivity of a substance.
The two sides of a metal plate 1.5 square metres in area, and
0.4 cm. in thickness are maintained at 100°C. and 30 D C. respectively.
If the thermal conductivity of the metal be 0.12 C.G.S. units find
the total amount of heat that will pass from one side to the other
in one hour. (Dacca, B.Sc.)
4. The interior of an iron steampipe, 2.5 cm. internal radius,
carries steam at 140 C C. and the thickness of the wall of the pipe is
3 mm. The coefficient, of emission of the exterior surface (heat lost
econd per sq. cm. per degree excess) is 0.0003 and the tempera
ture of the external air is 20 C C. If the thermal conductivity of iron
is 0.17 C. G. S. unit find the temperature of the exterior surface, and
bow much steam is condensed per hour per metre length of tube,
a tent heat of steam at 140°C, being 509. (London, B.Sc.)
5. The thickness of the ice on a lake is 5 cm. and the tempe
of the air is  10°C. At what rate is the thickness of the ice
dmately how long will it take for the thickness
■ oubkd ?
(T ■; conductivity of ice == 0.004 cal. cm 1 sec* °C~\ Density
of ice — 0.92 gm. per c.c' Latent heat of ice = 80 cal. per gm.).
6. Distinguish between thermal conductivity and therm ometric
conductivity of a substance. Describe a method of finding the
thermal conductivity of a solid.
Calculate the rale of increment of die thickness of ice layer on
a lake when the thickness of ice is 20 cm. and the air temperature
is 40°C. Thermal conductivity of ice = 0.004 cal. cm. 1
"^C 1 , density o£ ice = 0.92 gm./c.c. and its latent heat = 80 caL/jgffl.
After what time the thickness will be doubled ? (Bihar Univ., 1954.)
7. Define the terms conductivity and diffusivity as used in the
thcory of heat conduction. Describe a method of comparing the con
ductivity of two metal bars. Account for the fact that the evapora
tion of liquid air is greatly reduced when kept in a Dewar vacuum
vessel. (Madras, B.Sc.)
8. Define conductivity. Deduce expression for the flow of heat
in a long; bar when it has acquired a steady state. (B. H. U., B.Sc,
1931J
EXAMVLliS
Ml
9. Define the coefficient of heat conductivity of a substance
and give details of some method of determining this constant for
iron.'"' (A. V., B.Sc. 1931.)
10. Describe lngenHausz's experiment, and prove from the
mathematical theory that the conductivities of different bars vary as
the square of the length up to which wax is melted. (A. U., B.Sc,
1945, 1950.)
IL Distinguish between thermal conductivity and thermometrie
conductivity Bring out, the connection between the two. Describe
a tie's method of determining the thermal conductivity of a solid.
(Punjab Univ., 1950.)
VI. Show that in the steady state of a metal bar heated at one
end
Ifr = ***
■where the symbols have their usual significance.
Hence prove that the length to which the wax melts in the steady
state along a wax coated bar is proportional to the square root of the
coefficient'of thermal conductivity of the material of the bar. (Allaha
bad Univ., 1955.)
13. Describe Forbes' method of determining the thermal con
ductivity of a metallic bar, and explain the formulas used. (Lucknow
Univ., 1950 ; Allahabad Univ., 1951 ; Punjab Univ., 1954, 1955, 1957 J
Utkal Univ., 1953.)
14. Define thermal conductivity, A steady stream of water
flowing at the rate of 50 grams a minute through a glass tube 30 cms.
long, 1 cm. in external diameter and 8 mm. in bore, the outside of
which is surrounded by steam at a pressure of 700 mm., is raised in
temperature from 20°C. to S0 a C. as it passes through the tube. Find
the conductivity of glass. You are given that log 1.25 = 0.223.
Deduce anv formula that you use. (A, IL, B.Sc, 1937 ; Patna Univ.,
1948.)
15. Define thermal conductivity. Describe a method you have
adopted in experimentally finding this constant for a good conductor.
Find the coefficient of conductivity of a badly conducting material
upon which, the following experiment was made ;— A very thin walled
hollow silver cylinder 40 cm. in diameter and 50 cm. in length is
covered all over its external surface including 1 the ends by a layer of
the material 0.33 cm. in thickness. Steam at a temperature of 100°C
is passed through the cylinder and the external temperature is 20°G
r is found to accumulate within the cylinder at the rate of 3 gm.
per minute. The latent heat of vaporization of water at 100°C. is
537 cal. per gram and crosssections of the steam inlet and outlet are
each 10 square centimetres. (Madras, B.Sc.)
16. Define conductivity and diffusivity of hcaL When steam
is passed through a circular tube of length I and having the internal
342
JCXAMPIJES
and external diameters a and b respectively, prove that the radial flow
of heat outwards is given by
2irKl((t x e 2 )/lag e ~
where K is conductivity and $ t , 0. 2 the temperatures inside and out
side the tube, How will you determine the conductivity of india
rubber? (A, U., B.Sc„ lions., 1930.)
17. Steam at 100°C. is passed through a rubber tube, 14.1? cm.
length of which is immersed in a copper calorimeter of thermal capa
city 23 cat, containing 440 gm. of water. The temperature of the water
and calorimeter is found to rise at the rate of 0.019 d C. every second
when they are at the room temperature (22 °C.) . The external and
the internal diameters of the tube are 1,00 cm, and 0,75 cm. respec
tively. Calculate the conductivity of indiarubber.
18. Prove Lhat if a long bar is periodically heated at one end,
the law of propagation of heat is given by the equation
dd _ _^0
" * iP '
Obtain a solution of this equation and show how with its aid the
diurnal and annual variation oh temperature at some depth below the
surface of the earth can be explained. (A, U., M.So, 1926.)
19. Heat is supplied to a slab of compressed cork 5 cm. thick
and of effective area 2 sq. metres, by a heating coil spread over its
ace. When the current in this coil is 1.18 amp. and the potential
difl s its ends 20 vol is, the steady temperatures of the
E the slab are 12.5°C. and 0°C. Assuming that the whole of
the heat developed in the coil is conducted through the slab, calculate
conductivity of the cork.
Define coefficient of thermal conductivity and describe Lees'
hod for determining the thermal conductivity of metals. Equal
jer and aluminium are welded end to end and lagged.
If i 1 ends of copper and aluminium are maintained at 100 o C
and 0°C respectively, find the temperature of the welded interface.
Assume the thermal conductivity of copper and aluminium to be
0.92 and 0.50 respectively. (Punjab Univ., 1953, 1956, I960.)
21. Describe Lees' method of determining the thermal conduc
tivity of a bad conductor, (Bombay Univ., 1947.)
. Describe and explain the cylindrical shell method of di
mining She conductivity of a solid. (Bombay Univ., 1948.)
23. Describe and give the theory of a method useful for a
practical determination of the thermal conductivity of a liquid.
(London.. B.Sc.)
24. Discuss some methods by which the thermal conductivity
of a gas has been determined. What are the experimental difficult.':
and how have they been overcome ?
25. How is the thermal conductivity of a liquid determined ?
EXAMPLES 3*3
How do you demonstrate that hydrogen is more conducting than
air? (A. U., B.Sc, IS
20, Discuss the difficulties which beset the investigation of the
thermal conductivity of gases and indicate how and to what extent
they have been overcome, (London, B.Sc.)
IX
I. What do you understand by a reversible cycle as opposed to
an irreversible one? Give instances of each, (A. U., B.Sc., 1931.)
What is meant by a reversible change? Describe Carnot's
cycle and prove that the efficiency of all reversible engines working
between the same two temperatures depends only on the temperature
of the hot and cold bodies. (Bombay Univ., 1947, 1948, 1949 ; Lkkal
Univ., 1954; Punjab Univ., 1953.)
3. Describe Carnot's cycle and prove Carnot's theorem. What
•nt by a reversible change ? State briefly how Carnot's theorem
h to an absolute scale. (Delhi Univ.., 1952, 1954).
4. State and explain the significance of the second law of thermo
dynamics. Show that the efficiency of a reversible engine is maximum.
iiv., 1950 ; Punjab Univ., 1957.)
Describe a Diesel engine and deduce an expression for its
efficiency. Can the Carnot engine be realised in practice? (Punjab
Univ., 1949.)
fi. Describe with diagrams an Otto engine and deduce an ex
ion for its efficiency. (A. U., B.Sc, 1949.)
7. Describe some kind of internal combustion engine and explain
fully how heat is thereby converted into work.
Describe its uses and applications and dwell upon its advantages
over the steam engine. (A. U., B.Sc.)
8. Describe the cyclical process of a steam engine and compare
its efficiency with that of an internal combustion engine. Explain
(1) why steam engine is preferred in railways, and (2) why petrol
engine is used in. aeroplanes. (A. U,, B.Sc, 1930.)
9. Write an essay on 'Heat engines'. (Dacca, B.Sc)
10. Explain the indicator diagram and apply it to Carnot's cycle.
Explain the conditions for reversible working and show that in ge
for reversible cycles 1 y=0. (Bombay, B.Sc)
I I . Describe Carnot's cycle. Show how the work done during
each operation is represented on a pv diagram. Find an expression
for the Work done during ouch operation when the working substance
is a perfect gas, (London, B.Sc.)
12. Discuss the statement : 'Reversibility is the criterion of
perfection in a heat engine'. Explain with two examples what vou
understand by a reversible process.
A Carnot engine works between the two temperatures I00°C. and
10*C. Calculate its efficiency. (A. U., B.Sc, 1947.)
344
EXAM
X
EXAMPLES
345
1. Show that when a body expands, the external work performed
is given by the expression
From the following data calculate what fraction of the specific
heat of copper is due to the external work done in expansion in an
atmosphere at a pressure of 76 cms. of mercurv :—
Specific heat of copper = 0.093, Specific gravity of copper = 8,8,
of mercury = 13. 6. Coefficient of linear expansion of copper = 0.00 On 1 6,
J = 4,2 X 10 7 ergs per calorie. (A. U., B.Sc)
2. Deduce an expression for the work required to compress
adiabatically a mass of gas initially at volume v x and pressure pi to
volume v 2 .
Find the work required to compress adiabatically I gm. of air
initially at N.T.P. to half its volume, Densitv of air at N.T.P,
=== 0.00129 gm./c.c. and c p jc =z 1.4. (Birm,, B.Sc.)
3. Explain external and interna! work. Discuss the changes
in the kinetic energy of the molecules of a gas when heated and
hence show that the ratio of the specific heats for a monatomic eai
is 5/3. *
4. Write an essay on the transformation into heat of other forms
of energy. (Cal, B.Sc.)
5. Tn what sense can the second law of thermodynamics be
ded as furnishing ail absolute scale of temperature ? How can
readings of gas therm i be reduced to this particular scale ?
!J.; B.Sc, Bom., 1931.)
i what you .i id by a thermodynamic scale of
rature. Show that it agrees math an ideal uas scale. (Punjab
Univ., 1053, 1954, 1956, 1957; Bombay Univ., 1953.)
7. Write notes on the following : —
(a) Lord Kelvin's absolute scale of temperature,
(b) Joule and Kelvin's porous plug experiment,
' B.Sc, 1930.)
8. How did Kelvin arrive at the absolute scale of temp
Show that the ideal gas scale and the absolute scale are Iden
[Iow is the absolute scale I in practice ? (Delhi Univ.,
(Nagpur,
ture ?
deal.
1951, 1953, 1956.)
9. Define a scale oF temperature without malting use of the
peculiarities of any selected thermometrit substance. Show (a) that
Kelvin's work scale is such a scale, and (b) that the ratio of two tem
peratures as measured on the Kelvin scale is identical with the ratio
of the same two temperatures on the perfect gas scale. (London,
10. Explain what you mean by die entropy of a substance.
Show that for any reversible cyclic change of a system the total change
of entropy is zero. Explain why this statement is not true for irrever
j changes. (Punjab Univ./ 1958; Baroda Univ., 1955.)
11. Explain the idea of entropy. Derive an expression for the
entropy of m grams of perfect gas. (Allahabad Univ., 1950.)
12. A volume of a gas expands isotherm ally to four times its
initial volume. Calculate" the change in its entropy in terms of the
gas constant. (Baroda Univ., 195C)
13. Calculate the change of entropy when 100 gms. of water at
S0 Q C. are mixed with 200 gms. of water at 0° assuming that the
specific heat of water is constant between these temperatures.
(London, B.Sc.)
14. Derive an expression for the entropy of a perfect gas in
terms of its pressure, volume and specific heats. (Bombay, B.Sc.)
15* Prove that the increase of entropy per unit increase of
volume under constant pressure is equal to the increase of pressure
per unit increase of temperature during an adiabatic change. (Bombay,
B.Sc.)
16. Calculate the change of entropy when 10 gm, of steam
at 100°C, cools to water at 0°, assuming that the latent heat of
vaporisation is 536 and the specific heat of" water is 1 at all tempera
tun.
17. State the second law of thermodynamics and apply it to
the determination of the effect of pressure on the melting point of a
solid. (A, U., B.Sc)
15. Give an elementary proof of Clapeyron's relation
dp L
dT 1 (r s — v L Y
Discuss how the boiling point of a liquid and the melting point
■a solid are affected by change of pressure. (Punjab Univ., 1917,
1957.)
19. Derive Clapeyron's equation
dp L _
dT 7{p 2 — v t ) '
Calculate the change in temperature of the boiling point of
water due to a change of pressure of 1 cm. of mercury. (L = 536
calories, volume of 1 gm. of water at 100°C = 1 c.c, volume of 1 gm.
or saturated steam at 100°C — 1600 c.c.) . (Delhi Univ., 1955.)
20. Prove the thermodynamic relation
>a\ tdp\
\ dv Id ' \dOL
and hence prove
ft'0
(**)
(B. H. U„ B.Sc, 1931.)
346
KXAMPi.KS
21. Deduce the latent heat equation
Calculate the depression of the melting point of ice per atmos
pheric increase oL" pressure, given latent heat of fusion — bO cal. and
njity of ice at 0°C. 10 ~, „ c ir iar.\
r nor , — rr (Nagpur, B,Sc„ 1930.)
density of water at O'C. 11 v bl l
22. Prove that
One gram of watervapour at 100°C, and atmospheric pressure occu
pies a volume of 1640 c.c. and L — 53G calories, Prove that the
vapour pressure of water at 99°C. is about 733 mm, of mercury.
(A, U., RlSc, 1926.)
23. If sulphur has a specific gravity 2.05 just before, and 1,95
just after melting, the melting point being 115°G. and the latent
heat 9.3, find the alteration in melting point per atmospheric eha
of pressure. (I.ond., B.Sc, Hons.)
24. Calculate the von I me of n gramme of steam at 100°C, given
that the lati i of evaporation of water at 10G°C = 535 and
the change of boiling point is 0,37°G. per cm. of mercury pressure
(/ £=4.2X10*). (Manch., B.5c.)
25. Calculate £ given that the change of
of one phere changes the melting point of ice by
C and when one gram of ice melts volume changes by Q.O907
(Punjab t
26. Discuss the effect of change of pressure on the polling point
of a liquid.
olum.es of water and saturated steam
at 1Q0°C are 1 c.c. and 1601 c.c. respectively and the latent heat
vaporisation is 536 cal./gm., find the change in boiling point for a
change of pressure of 1 cm. of mercury. (Allahabad Univ., 1946 j
Calcutta Univ., 1948.)
27. Show, on thermodynamic principles, that when a liquid
film is suddenly expanded, it must fall in temperature, and find an
expression for the lieat that must be supplied to keep it constant.
(A. U., B.Sc, Hons., 1927.)
28. Deduce an expression for the specific heat oi saturated
vapour and prove that in the case of water at 100°C., It is negative.
How do you explain the paradox 'negative specific heat ?' (A. U.,
B.Sc., Hons., 1931.)
29. Define the 'triple point*. Describe the successive changes
observed in a system containing water at its various states when
pressure is changed at a constant temperature (a) when the tern
■e is above the triple point, (b) when the temperature is bel<
it. (Dacca. B.Sc., 1930.)
EXAMPLES
30. The latent heat of steam at 100°C. is 536, calculate what
fraction of this heat is used up in performing external work during
vaporisation, assuming the density of steam at 100°C to be 0,007
and an atmosphere to be 10 c dynes per sq. cm. (Man eh, B.Sc.)
31. Define the 'triple point' and show that the steam line, hoar
frost line and the iceline must meet in a single point. Draw the
isodiennals of water for temperatures lower than that of the triple
point. (A. U., B.Sc, 1934.)
$2. Define Entropy. What is its physical significance? Show
that the entropy remains constant in a reversible process but In
creases in an irreversible one. (A. U., B.Sc., 1936.)
S3. Define Entropy. Prove from the principles of thermody
namics that the decrease of entropy per unit increase of pressure
during an isothermal transformation is equal to the increase of volume
per unit increase of temperature under constant pressure.
Hence show that heat is generated when a substance, which
expands on heating, is compressed (A. U., B<5c, 1938.)
34. What do you understand by 'entropy' ? State the second
law of thermodynamics in terms of entropy.
Calculate the change in entropy when m grams of a liquid of
specific heat s are heated from 2V to r 2 °, and" then converted into
its vapour without raising its temperature. [Latent heat at 7V =L.]
(A. IP, B.Sc,, 1941.)
35. Prove the latent equation
dL L
dT T ~ H H
Calculate the specific heat of saturated steam from the above equa
tion and explain the meaning of its negative: value. Given L = 539.3
cal., T = 100°C ; d h, =  0.640, c t — 1.01. (Allahabad Univ., 1955 ;
at
Baroda Univ., 1953 ; Nagpur Univ., 1956.)
XT
1. State Newton's law of cooling. A copper calorimeter weigh
ing 15 gms. is filled first with water and then with a liquid. The
times taken in the two cases to cool from 65°C to 60°C. are 1 70
sec. and 150 sec. respectively. The weight of the water is 11 gms.
and that of the liquid is 13 gms. Calculate the specific heat, of the
liquid. The specific heat of copper ia 0.1. (Cal., B.Sc.)
2. State Stefan's law and discuss in the light of the same (I)
Newton's law of cooling, (2) the temperature of the sun, and (3) the
perature of a tungsten arc. (A. U., B.Sc, 1932.)
3. State and deduce KirchhofFs law on the emission and absorp
tion of thermal radiation. (Sheff,, B.Sc.)
4. What is a black body ? What are the characteristics of a
348
EXAMPLES
blackbody radiation? How has it been realised In practice? Des
cribe how Stefan's law has been verified w T ich it. (Allahabad Univ.,
1949; Punjab Univ., 1961.)
5. How can you show that the radiation from an enclosure
depends only on the temperature and not on the materials of its
walls.
Describe a radiation pyrometer, (Delhi Univ., 1955.)
6. Explain the terms 'emissive power" and 'absorptive power'.
Deduce that at any temperature the ratio of the 'emissive power to the
absorptive power 'of a substance is constant and is equal to the
emissive power of a perfectly black body. (Baroda Univ., 1951.)
7. Discuss the evidence both theoretical and experimental show*
ing that good emitters are good absorbers. (Punjab Univ., 1953.)
8. Explain what you understand by a black body. State
Stefan's law of radiation and prove, it from thermodynamkal consi
deration. Indicate how it can be verified. (Punjab Univ., 1944,
1950; Roorkee Univ., 1951, 1959.)
9. Explain what is meant by a "perfectly black body". Write
a short account of the distribution of energy in the spectrum of the
radiation from such a body. (Leeds, BJSe.)
10. Describe any furnace and explain how you would measure
its temperature, (A. U„ B.Sc)
11. Discuss the principles underlying the measurement of tern
inriinT pyrometers, and show how the value
of the temperature is estimated. Describe fully and clearly a prac
tical form of apparatus of this type. (A. U., Br.Sc, lions., 1930.)
12. DeOne solar constant. Explain with necessary theory how
the solar constant, is determined. How is the temperature of the
estimated from the data of the solar constant ? (Allahabad Univ.,
i Punjab Univ., 1952; Nagpur Univ., 1953.)
13. Under what conditions may a thermodynamic investigation
of the radiation within an enclosure be carried out?
Show that under these conditions the radiation varies as the
fourth power of the temperature, (A. U„ B.Sc*, Hons,, 1928.)
14. Describe the construction of Boys' radiomicrometer and
explain the principles involved in its action. (London, B,Sc)
15. Calculate the temperature of the Earth, assuming that it
absorbs half the energy falling on it from the sun and that the sun
radiates as a black body, (Radius of Sun 7x 10 10 cm ; radius of
Earth — 0,3 >( 10 s cm; mean radius of orbit of Earth = 1,5 X 1& 18 cm J
temperature of sun's surface— 6Q00°K; SteFan's constant <rz= 5,7 X 10 8
C.G.S. units,)
16. State StefanBoltzmann law of radiation and describe
briefly experiments by which it has been confirmed. (A. U., B,Sc„
1934.)
EXAMPLES
349
(in)
(y)
(vi)
(vii)
17. What is a perfectly black body ? How can such a body be
realised in practice ?
How will you verify experimentally that the ratio of the emissive
and absorptive powers k the same for all bodies and is equal to the
emissive power of a perfectly black body? (A. U., B.Sc, 1940.)
18. State Newton's law of cooling mentioning its limitations.
How would you verify it experimentally ?
\ body initially at 8D°C cools to 64°C in 5 minutes and to 52°<
in 20 minutes. What will be its temperature after 16 minutes and
what is the temperature of its surroundings? (Punjab Univ., 19oI,
1952 ; Patna Univ., 1951,)
19. Describe the radiation method of measuring high tempe
ratures. (Delhi Univ., 1951.)
20. Write short notes on the following :—
(i) Prevost's theory ot exchange. (Punjab Univ., 195 1.) ,
Perfectly black body and its actual realisation. (Delhi
Univ., 1952,)
Blackbody radiation. (Delhi Univ., 195G.)
Stefan's law. (Delhi Univ., 1955,)
Solar constant. (Delhi Univ., 1956.)
Determination of the temperature of the sun, (Delhi
Univ., 1952.)
Optical pyrometers. (Allahabad Univ.)
XII
1. What is meant by (a) relative humidity, (b) dewpoint ?
Describe the Rcgnault's hygrometer and explain how relative
humidity can be measured with its help.
2, Explain the general principles underlying the use of a wet
and dry bulb hygrometer for determining the hygrometric state of
the atmosphere.
3. Describe the hair hygrometer and state its uses.
4, Assuming convective equilibrium of the troposphere, find
an expression for the decrease in temperature as we go upwards in
the atmosphere.
ANSWERS TO NUMERICAL EXAMPLES
2, 50°C.
I
3. 1.55:1.
4. 1.29 X 10» ei j
II
2. 822 min. nearly,
G. 4.217 X ™ 7 ergs/cal
9. $.89 cal. per gm. per °C.
10. (a) ; (b) 412.8°C,
11. )5.9°C.
13. 51,8°C
17. 228°C,
20. 1.60.
21. 1.402
350
EXAMPLES
III
1. 4.22 >< 10 T ergs.
3. 1.91 X 10° cais.
5. 4,16 x 10 T ergs.
7. 40.5 cat,
II. 3.47 X 10* cro./sec,
15. 5.67X I" 10 ergs.
2, 29,5 cal.
i 9.35 h.p. ; 13.1 °C. per min.
6. 4.116 X 10 7 ergs.
10. 0.119°C.
14. 4.66 X 10* cm./sec.
IV
3. 33,6 °C. {a arid h are given in atmospheres and c.c. respec
tively for 1 ex. of gas at N.T.P.).
4. 100.5 cal. per gm.
V
VII
2. 475X10*.
t. 59.6 sec.
10. 0.000801 sq. cm.
3. 5.646 sec. per day gain per °C. rise,
6. 0.8 X 10 4 per °C. 7. ■ JL
II. 1.20:1
VIII
3. 11.3 X 1 8 cal.
0.39 cm. per hour; 19 hours
40 minutes.
14. 1.31 X 10 3 * 15. 1.26 X 10 B C.G.S. units,
3.54 X 10~*. 1 !i, I . ] 3 x 10* C.G.S. units,
20. 64.8°C
X
i.
6.28 X 10 s ergs. 12, 1.385JJ. 13. 0.36 unite,
21. O.OOSPC, 23. 0.0252°C. 24. 1674 c.c
25. 80,8 calories. 26, 0.36°C, 30. 35. 1.07.
1
19. 0.36*
XI
1. 0.733.
15. 290°K,
18. 43°C : 16°a
The Gas Constant
Coefficient of expansion for
perfect gas at 0°C.
Ice point
Volume of one grammolecule
at N, T. P. =
Avogadro number N =
Loschmidt number n =
Mass of H 1 atom M =
Standard gravity g s=
Density of mercury at N.T.P. =
Standard atmosphere s=
Mechanical equivalent of I
Boltzmann's constant k =
StefanBoltzmann constant a—
Wien's constant b ==
Planck's constant k :=
Velocity of light in vacuum
Electronic charge <■ ==r
Faraday number
Mass of the electron m
ific electronic charge e/m
Gravitation constant
PHYSICAL CONSTANTS*
R = (8.31436 ±0.00038) X 10 T ergs.
deg 1 mole 1
= 1.98646 =h 0.00021 cal. deg 1
mole— 1 .
0.0036608 per °C.
■273.16 ihO.OPK.
; 22.414 litres
: (6.0228 ± 0,0011) X 10 28 mole 1
(2.6870 =fc 0.0005) X 10 ,ft cm,''
(I.67S39 ± 0.00031) X 10" 2 ' 1 gm*
980.665 cm. sec. 2
13,59504 ± 0.00005 gm. cm*
1,013246 X 10* dynes cm. 2
,i 04) X ™ 7 ergs cal. 1
(1.38047 ± 0.00026) X 10" ergs.
(5.672 ± 0.003) X 10" 6 erg. cm 2
deg." 1 sec. 1
0.28971 ± 0.00007 cm. degree
(6.624 ±. 0.002) x 10 ~ 2V erg, sec.
(2,99776 ± 0.00004) X 10 10 cm.
c — (2.99776 ± 0.0010) X *0 1( > e.s.u.
= (1.60203 =b 0.00034) X 10~ 20
e.m.u.
V = 96501 dr 10 int. coul. per gm.
equiv.
(9.1066 ± 0.0032) X 1() ~ 2B gm.
) xlO 7 e, m. n.
per gm.
G = (6.670 ± 0.005) X lO 8 dyne cm. 2
em. 3
* The values given here are taken from Birue, Reviews of Modern
s, Vol. 13, p. 233 (1941) ; Amer. Jour! Phys., Vol. IS, p. 63
5). All quantities in this table involving the mole or the gram
equivalent are on the chemical scale of atomic weight (O = 16.0000) .
351
/
SUBJECT INDEX.
The numbers refer to Pages
Absolute expansion of liquids, 168
ilute null point, see Absolute zero,
lute pyrheliomcter, 312
Absolute scale of tempera ture, 249
Absolute zero, 5, 2S0
Absorption coefficient, 286
Absorption freezing machines, 127, 131
Absorption of radiation, 28fi, 289
Absorptive power, 286, 289
:.Ljc change of humid air, 325
Adiabatic demagnetisation, 147
— equilibrium, 318
Adiabatic expansion method
ing % 53
Adiabatic expansion of compressed
gases, 132
Adiabatic stretching of wires, 265
Adiabatic transformations, 4R
Adsorption, 125, 133
Air ci : 151 et .
— machine, 154
151
liquefacti ictton
air.
mtial, 2?8
1 1:j
its, 97
machines, 131
machines,
Amor lids, 105
Andrews' experiments ■ ctivity
of gases, 198
Andrews' experiments on carbon
dioxide, 88
s experiments, 187
isotropic bodies, thermal expansion
of, 165
application of Klrcbhoff's
law to, 291
Athermancy, 288
Atmospberej thermodynamics of, 316
pi seq.
— , distribution of temperature in, 316,
319
— , distribution of pressure in, 318
— t water vapour in, 319
ispherfc engine, 206
Atomic energy, 79
Atomic heat, 43
— , variation with temperature, 45
Available energy, 243,
Ige velocity, 78
Avogaiiro's law, 75
— number, 76
B
Bartoli's proof of radiation pressure,,
301
Baths, fixed temperature, 11
— . sulphur, 12
Bcckmann's thermometer, 3
Becquerel effect, 21
BellColeman refrigerator, 133
Berthelot's apparatus, 118
Bimetallic thermorcgulator, 174
Blackbody,
— , absorptive power of, 286, 290
— , definition of, 286
— , : i —wer of, 290
— of Fery, 297
— oi Wien, 296
— , spectrum, emitted by, 305
— , total radiation from, 302
Blackbody curves, 305
Blackbody radiation, 295, 302
Blackbody temperature, 308
■ — , measurement of, 307 et seq.
— of tlie sun, 314
point of water, 11
— , variation with pressure of, 263
Bolometer, 299
— , linear 299
— , surface, 299
Bo nb calorimeter, 62
Boyle's law, 5
— , deduction from kinetic theory, 75
— >, deviations from, 87
Bridges for resistance thermometer, 15
Broadcasting waves, 284
Brownian movement, 71
Buusen'iH calorimeter, 33
Caikndar and Griffiths' bridge, 16
Callendar's continuous flow calorimeter,
38
Caloric theory, 64
Calorie, definition
Calorimeter,
— , bomb, 62
— , Bunseii's see, 33
— j continuous flow, 38
— j copper block, 31
— , differential steam, 36
— .i July's steam, 34
— , Nernst vacuum, 41
— , steadyflow, ■■'..
Calfcrimetry, Chapter II
— , electrical methods, 37 et seq,
— , method oi cooling, 32
— , method of mixtures, 29
— , methods based on change of state,
33
Carbon dioxide, critical constants of,
90, 102
— , isothermal curves of, 89
— , theoretical curves for, 93
Caruot's cycle, 213 *f seq.
i hot's engine, 213 et seq.
. reversibility of, 218
; theorem, 218
Cascade process of refrigeration, 135
Change of state, Chapter V
— , application of thermodynamics to,
262, 272
Charles' law, 5, 176
Chemical thermometers, 3
Claude's air liqucfier, 143
ClaudeHeylandt system, 144
,. ClausiusOapeyron equation, 262
— , Clapeyron's deduction of, 272
Clement and Desormes* apparatus, 53
leal thermometer, 3
Coefficient of performance, 235
Combustion engines, see Internal com
bustion engines
Comfort chart, 151
Comfort zone, 152
Comparator method, 159
Compensation of clocks and watches,
172
Compensation of mercury pendulum,
171
Compound engine, 212
idensation, methods of causing, 324
Conduction of heat, Chapter VIII
— , in three dimensions, 191
— r Kinetic Theory of, 84
— through composite walls, 190
Conductivity, thermal, definition of,
179
— . of different kinds of matter, 178
23
, relation between electrical
tivity and, 190
Conductivity of Earth's crust, 188
Conductivity of gases, S4, 197 ei seq.
— , relation between, viscosity and, 84
— , variation with pressure, 84, 199
Conductivity of glass, 194
Conductivity (thermal) of liquids, 196
et seq.
Conductivity (thermal) of metals,
determination, 180 et seq.
— , by a combination of steady and
variable flow, 186
— from calorimetric measurement, 180
— from periodic flow of heat, 187
— from temperature measurements, 182
Conductivity of poorly conducting
solids, 192 et seq.
Conductivity of rubber, 194
Conductivity, thermometric, 182
Constant, Boltzmann, 76
— , critical, see Critical constants
— , gas, 6, 10
— , Planck's, 304, 351
— , solar, 311
— t Stefan's, 302
— , Wien's, 304, 306, 351
Constant flow method for gases, SI
— for liquids, 38
Constants in kinetic theory, 85
Continuity of spectrum, 281
Continuity of states, 90
Convection of heat, 201
— , forced, 203
— , natural, 202
Convective equilibrium, 318
Cooling by adiabation expansion, 132
— due to desorption, 133
— due to JouleThomson effect, 136
et seq.
— due to Peltier effect, 133
— , Ncwton'slaw of, 287
— , regenerative, 141
Correction for emergent column, 171
Correction of barometric reading, 170
Correction of gas thermometer from
JouleThomson effect, 269
Crank, 210
Critical coefficients, 96
Critical constants, 90
— , deduction from van der Waals'
■:viriti;,n, •'.!.'<
— , determination of, from law of recti
linear diameters, 100
— , tabic of, 102
Critical density, 101
subject iNnnx
355
Critical point, 90
— ., matter near the, 102
Critical pi i 95, 100
— temperature, 90, 95, 100
— volume, 90, 95, 100
CryophortiK, 126
Cryustatg, 149
Crystals, expansion of, 165
Cycle, Carfiot, 214
— — , efficiency of, 216
—j reversibility of I
— , Diesel, 226
— »— , efficiency of, 227
— Otto, 223
, efficiency of, 224
—, Rankine's, 220
, efficiency of, 221
Cylindrical shell method o finding
CI iiiuCtivLLy, 193
I J
Dalton's law of partial pressures, 75
■ ty lamp,
Dead space correction, 9
.7''
■,'apour, m Vapour density
• from I ! ■;■ Ic's law,
— sei
Die
: air ili: • . 278
Diffuse radiati n, energy density of, 301
— , pressure of, 301
meter method, lci7
Disappearii ient pyrometer, 310
: ir poor conductors, 194
tti energy, 245
Displacement law, Wien's, 304
Distribution of energy in blackbody
rum,
Distril (Maxwell .
ting engine, 208
ie's theory of conduction, 191
Dslide valve, 209
Duiong and Fctit's law, 43, 81
— , from kinetic standpaint, 81
— , illustration of, ii
Dynamic method of finding vapour
pressure, 114
Earth, temperature inside the, 188
Earth's crust, conductivity of, 188
Eccentric, 21 1
Effective temperature, 153
Efficiency of engines, 212
— , Camot's cycle, 216
— , Diesel cycle,
— , Otto cycle, 225
l\]i.i.:trical methods in Calorimetry, 37
et seq.
nprir "magnetic waves 283*234
Emission coefficient, 290
— , total, 302
Emissive power, 290
Energy, conservation of, 242
— , chscviiuiHHMK changes in, 245
— , dissipation of, 343
— , distribution of, in the spectrum, see
Distribution of energy in the black
body spectrum
— , forms of, 241
— , molecular and atomic, 79
— , transmutation of. 242
Engine, Carnot's, 213
— , Diesel, 226
— , hotbulb, 228
— , Internal combustion, see internal
combustion en
— , jet, 234
— , National gas, 228
— , OtitO, 223
— i re ; irreversible, 218
— , semiDiesel or hotbull i.
, singleacting, 209
Entropy, 25 1 et seq.
rgy, 256
— , change of, in reversible pp ■ 253
— — , in irreversible pr icesse . 254
i definition of, 251
— , law of increase of, 255
— .if a perfect gas, 2~?
— of u system, 253
— of steam, 258
— , physical concept of, 257
— , statement of sec i thermo
dynamics, ill t< rn  of,
Entropytemperature dlagramj 257
3 T .ii::iiii:n of ClausiusClapeyron. 262,
— of heat conduction (Fourier), 183
Equati ms of state i p, IV
— , defmitirn of, 87
Kperimental study of, 96
— of van der Waals, 'Jl
Equilibrium, adiabatic or convective, 318
— , radiative, 317
Equipartkion of energy, 79
Eutcctic mixture, 126
— temperature.
Examples on Thermodynamics, 271
Exchanges, Frevosfa theory of, 286
Expansion (thermal). Chap. VI f
— , applications of, 172174
Expansion of anisotropic bodies, 165
Expansion of crystals, 161, 165
— j, Fizeau's method, 161
— , fringewidth dilatometer methodj
163
Expansion of gases, 175
— , determination of, volume co
efficient of. 175
Expansion of invar, 165
i 1 .  ' •; i.i of liquids, 166 et seq,
— , absolute, by hydrostatic balance
method, 169
— , dilatometer method, 167
msion of silica, 165
Expansion (linear) of solids (isotro
pic), 157
— , discussion ::i results, 164
— , earlier measurements of, 158
— , measurement of, 159 cl seq.
, comparator method, 159
■, J .;•:■■ titer and Laplace's method,
159
— relative, hy Helming r S tube method,
160
Expansion of water, 171
Expansion, surface and volume, lb5
P
Film method of determining conducti
vity of liquids, 196
— of gases, 198
First law of thermodynamics, see Ther
modynamics, first law of
Fixed points, chart of, 2526
Fixed temperature baths, 11
Flow of heat (rectilinear), 182
— , combination of steady and variable,
18! i
— , periodic, 187
■heel, 210
ies' method, 186
unhofer lines, 291
— spectrum of the sun, 291
Freedom, degrees of, 79
Free path, 81
— , mean, see Mean free path
Freezing mixture, 126
Freezing point, see Melting point
Frigidairc, 130
Fuel in engines, 229
Fusion, 104 ei seq.
— , effect of pressure on, 109
— — , thermodynamic explanation of,
262
— , latent heat of, see Latent heat of
fusion
 of alloys, 110
Gamma rays, 285
Gas engine. 332 1 1 seq.
Gas laws. 5
— , deduction of, from kinetic theory,
7B
— , deviation of gases from, 87 et seq.
perfect, 9, 87
Gas scale corrections from Joule
Thomson effect, 269
Gas scale, perfect, 9
Gas thermometers, 4
— , Calendar's compensated air thermo
meter, 6
— , constant pressure, 6
— , constant volume, 6, 7
— , constant volume hydrogen, 7
— , standard, 7
Gases, conductivity of. 1 r 7
— , equation of state for, Chap, IV
— , expansion of, Chap. I & VII
— , liquefaction of, 133 et seq.
Gases, permanent, 194
Gases, specific heat of, 49 et seq. See
also Specific heat of gases
— , thermal conductivity of, 84, 197
■ — , viscosity of, 83
Glacier motion, 110
j, conductivity of, 194
Governor, 210
Gruneisen's law, 165
Guard ring, 181
H
Hair hygrometer, 323
Hampson's air liquefier, 143
Heat, a kind of motion, 64, 70
— and light. 281
— and work, 65, 243
— as motion of molecules, 70
— balance in human
81 BJECT 1NIHCX
Heat by friction, 6J
Heat, caloric theory of, 64
— , convection of, 201
— , dynamical equivalent of, 6S70
— engines, Chap, IX, see also Hngines
and Cycles
— , latent, 104, 107, 117
— — , variation with temperature, 120
— , nature of, 64 c! scq.
— of combustion, 62
— , periodic flow of, 187 at teq,
— , propagation of, J 78
— , radiant, Chap, XI, see also
Radiation.
— , rectilinear flow of, 182
— , specific, see Specific heat
— , steady flow of, 183
— t unit of, 28
Helium, liquefaction of, 146
— , solidification of, 147
Henning's tube method, 100
Hertzian waves, 282
Hess's law, 246
Hotbulb engine, 228
JlnLwire method of finding conducti
vity, 197, m
Human body, heat balance in, 63
Humid air, adiabatic change of
Humidity, 320
— , absolute, 320
— , relative, 320
Hydrogen, conductivity of, 198
— , critical temperature
— , JouleThomson cooling in, 144, 268
faction of, 144
— spectrum, 292
i crmometer, 7
Hydrostatic balancv 169
Hygrometer, 321
— , chemical, 321
— , dewpoint, 322
— , hair, 323
— , Regnauit's dewpoint,
— , wet and dry bulb, 3 J 3
Hygromctry, 32]
Hypsometer, 12
Ice, latent heat of, 105, 107
Indicator diagram, 213
Infrared ray?, 282
IngenHausz's experiment, 184
Integrating factor, 240, 260
Internal combustion engines, 222 el seq.
, application of, 229
— , fuel used in, 229
• i i * a gas h
panding, 47
i Ltional temperature cale, 24
Intrinsic energy, vnrin: , with
volume, 266
Invar, expansion of, 165
Inversion temperature, I
— , expression for, from thermo
dynamics, 269
Irreversible engines, 218
— process, 216
Isothermais, 89, 98
J
J, ili.:'.i.:niiin;ilii..i:. of, 66 et scq.
Jena glass, 2
Jet propulsion, 234
Joly's steam calorimeter, 34
— , differential form, 36
Joule's experiments. 47, 65
— law, 47
— method of finding y, 54
JouleThomson effect, 336. 268
— , Correction of gas thermometers
from, 269
— for gas obeying van der Waals'
— inversion of, 145
K
Kapitza's tiquefier, 144, 147
Kinetic theory of matter, Chap, II I
— , constants, table of, 85
— , deduction of gas laws from, 75
— , evidence of the molecular agitation,
71
— , growth of, 70
— , introduction of temperature into, 76
— of specific heat, 7981
— , pressure of perfect gas from, 72
Kirchhofl's explanation of FraunhofLr
lines, 292
Kirchhoff's law, 290
— , application to Astrophysics, 291
. deduction of, 2<>5
Langlcy's bolometer, 299
Laplace's formula, 3 If!
Lapserale, 316
Latent heat of fusion, 104
— , determination of, 107 et seq.
247
' m Lit. d of finding conductivity of
ii !■;, 1%
• ii poor conductors,
• I h , 278
ion by heating, 281
l ,in 1 1 ■'■■ ii achine, 142
i [near ton, 157 et scq.
ion of air, 142
Liquefaction of gases, 133, el
— by application of Joule Thomson
effect, 136 et seq
 by application of pressure and low
temperature, 133
— by cascades or scries refrigeration,
135
Liquefaction of helium by IC O lines,
146
Liquefaction of hydrogen, 144
Liquid air, uses of, 149
Liquids, expansion of, see Expansion
of liquids
— , thermal conductivity of, tit Con
ivity of li
Liquid thermometers, 14, 23
of planetary atmosphere, 78
Low temperature siphons, 149
Low temperature techniques, 148
Low temperature thermometry, 22
M
Matter, continuity of liquid mid gaseous
states, 90
— , state of, near the critical point, 102
Hatter, three states of, 104
Maximum and minimum thermometer,
3
Maxwell's demon, 256
Maxwell's law, graphical rcpreseuta
on of, 77
— of distribution of velocity, 76
Maxwell's tbermodynamical relation
ships, 260 et seq.
Mayer's hypothesis, 47
Mean free path, 81
— of Maxwell, 82 _
Mean square velocity, 78
Mean velocity, 78
Median ir tifvalent of heat, 6fi et
seq.
— , value of, 70
Melting, d electrical resistance
on, 106
— , change of vapour pressure on, 106
— , chat • volume on, 106
Melting point, 106
— of ice, effect of pressure on, 109,
262
— of metals, 106
, effect of pressure on, 263
Mercury, conductivity of, 181
— , expansion of, 170
Mercury thermometer
— , errors oi,
, correction for,
Method, electrical, of measuring specific
heat, 37 et seq,
— of finding latent heat, 108
— of cooling, 32
— of melting
— of mixtures, 29, 49
ant! latent heat of fusi'
107
— — and latent heat of fusion of
metals, 108
Methods of causing condensation.
Molecular I .mi, 44, 79
Molecules, diameter of, 81
— , free path of, see Mean free path
— , translation;)! energy of, in gas, 76
— > velocity of, in pas, see Velod
Multiple expansion engine, 212
K
National gas engine, 228
Natur> :, 64
Ner nst's copper block calorimeter, 31
Mernst's vacuum calorimeter, 41
Neumann's law of molecular heats, 44
Newton's law of cooling, 29, 287
Nozzle, expansion through, 230
Optical method of measuring expan
sion, 161164
Optical pyromctry, 309 see also Radia
tion pyrometry
Otto cycle, 223
Otto engine, 223, 229
Path, see Mean free path
Peltier effect, 21
— , cooling due to. 133
Pendulum, gridiron, 173
— , mercury, 171
$58
SUBJECT INDEX
Perfect differential, 239
Perfect gas, 5, 87
— t pressure of, 72
— scale, 9
Periodic flow of li.;it, 187
Permanent gases, 134
— , liquefaction of, see Liquefaction nf
Perpetual motion
— of first kind, 219,
— of second kind, 219, 248
Fetrolei m ether thermometer, 23
Phenomena, mean free path, 81
— , transport, S2
Phenomenon of conduction. 178 (Chan.
VIII)
— of viscosity, S3
Photosphere, 314
— , temperature of, 314
Planck's radiation formula, 304
Planet, artificial, 23S
Planetary atmosphere, loss of, 78
Platinum resistance thermometer, 1318
Poro. :periment, 139
Potentiometer, 20
p. 97
Pre::. ' 7n
30©
see
ure
Priji
Principle of regenerative 141
iPyrhel I lute, 312
— , v,
— , wi tir, 312
Pyrometry, 306 si
— » fPs pyrometers, 306
— , optical, 309, see also Radiation
pyp i i
— , radiation, 307
— , resistance, 307
— , thermoelectric, 307
Quantity of heat, 28
Quartz, 165
R
iant energy, identity of light and,
281 e! seq.
— , nature of, 279
— , passage through matter of, 28fi, 288
— , properties of, 279
Radiatioi . I . a XI
— , application of thermodynamics to,
see Therm dynamics of radiation
— , blackbody, 293297
, analogy between perfect gas and,
301
Radiation constant, Planck's, 304
— , Stefan's, 302
— , Wien's. 304
Radiation correction, 29
Radiation, diffuse, 301
— from the stars, 314
Radiation laws, Planck's law, 304
— , StefanBoltzmanft's law, ii
Radiation, measurement of, 297 et seq.
Radiation, passage through matter of.
Radiation, pressure of, 300
— , properties and nature of, 279
— , temperature, 293
Radiation pyrometers, 307 et seq,
liative equilibrium, 317
Radiometer, Crookcs', 298
i meters, 297 ei seq,
Radiomicrometcr, 300
220
', 221
i of the sped
.■rq.
—, adiabatic expansion method (Cle
ment and Desormes) , 53
— , experiments of Partington, SS
— , method of Ruchardt, 56
— , table of results, 62
— , velocity of sound method, 57
Reaction turbines, 233
Rectilinear diameters, law of, 100
Rectilinear flow of heat, 182
Reflecting power, 288
128
— , characteristics of, 129
Refrigerating machine, absorption, 131
— s air compression, 133
— BellColeman, 133
— , Carnol, 218
, efficiency of, 235
— , vapour compression, 127
——j efficiency of, 235
Refrigeration, Chap. VI
— , cascade process of, 135
— due to Pettier effect, 133
— , principles used in, 125
by adding a sal , 125
b i • der n duced
i VH ii:i,
 by Joule Thomson expo
Refrigci mi Refrigerating machine
i : u i . i '.••: OOOltng, 141
. thermal, 179
:e thermometry, 13, 23, ■'<'•'■'
, platinum, 13
engines, 218
— process, 216
cket, 234
Room 154
i mean square velocity, 74
Rowland's experiments, 66
n I icperiments, 56
..Ivc, 206
Satellites artificial, 2
Saturated vapour, 111
— , density of, 123
— , specific heat of, 273
— , vapour pressure of, 112117
Searle's apparatus for conductivity,
ISO
Secondary thermometers, 10, 27
. 228
Singleacting engine, 209
Slide valve, 209
Solar constant, definition of, 311
— , determination of, by absolute pyr
imeter, 312
Solidification of helium, 147
Solids, conductivity of, see Conducti
vity of solids
Sound, velocity of, 57 et seq.
Specific beat, definition of,
— , difference between the two, 46, 245,
— , l:!n ■ ''
 ~, methods of measurement. Chap, II
— , negative, 274
Specific heats of gases, 45 et seq.
Specific: heat of gases, determ'nati n
(experimental) at constant pressure,
49 et seq.
— , at constant volume, 51 et seq.
■ ! cory
Specifn heat of liquids, 31, 3740
d, 37
— , method of mixtures, 29
— , stead j flow electric calorimeter, 38
heat of vapour, saturated, 273
— unsaturated, 61
Specific heat of solids, 30, 41 et seq.
— , electrical method, 4143
— v method of mixtures, 29
— i variation with temperature.
Specific heat of superheated vapour, 61
• , 40
Spectra of stars, 292
. V
— , therm idynamic treatment of, 262,
272
States of .104
Steam calorimeter (Joly), 34 et seq.
, 205 ei sr ;
— , < 208
— i, modern, 211
— , New men's atnn spl &rj
arts of, 208 ei .
— , Savi py's, 206
— , singleacting, 209
Steam, enti
— , expansive use of, 209
— total beat of
ii jet, theory of, 230
, 229
mp ,23;
uf Curtis, 232
— of Dc Laval, 231
— of Parsons, 233
— , reaction, 2
StefatiiBoltzmaiui'a lav,, 302
— , experimental verification of, 302
Stefan's constant, : :
Stral here, 317
— , pressure did : n in, 318
Stuffing box, 209
Sublimation, 105
Sulphur boiling apparatus, 12
Sun, radiation from, 311 et seq.
— , temperature of i
iling, 111
h a ting, 111
, 61
259
360
SUJJJIiCT INDEX
Temperature, absolute, 10, 249
— baths, 11
, critical, 90, 95, 100
— — , table of, 136
Temperature, critical, value_ of, from
ve.jj tlur Waals' equation, 95
— , effective, 153
— , definition of, 1
— , distribution of, in the atmosphere,
316
_ gradient, 319
— , high, measurement of, 306
— , low, measurement of, 22
„ — t production of, Chap. VI
— of inversion, 145, 268
— of stars, 314
— of sun, 314
— radiation, 293
— scale, international, 24
— . standard, 11
— , underground, 188
— wave, wavelength of, 188
Theorem, Carnot's, 218
Theory, atomic, 70
— , kinetic, of matter, Chap. Ill
— , molecular, 70
— of exchanges (Frevost's), 286
Thermal definition of, 23
Xhet wt Condu.
(the i"' . ,
— , ratio of electrical conductivity to.,
190
Thermal exp slon
banco, 179
Thermal state of a bod: i
rmoenuples, 18 et se<"!. _
Thermodynamic relationships (3
well), 260 ei
— , first relation, 2
, application to change of freez
ing' point by pressure, 262
} application to liquid film, 264
_ second relation, 264
— , third and fourth relations, 266
Thermodynamic scale of temperature,
10, 249
Thermodynamica! variable, 237
Thermodynamics, application ot, to
change of state, 262, 272
Thermodynamics, application of, to
radiation, see Thermodynamics of
radiation
Thermodynamics, examples on, 271^
Thermodynamics, first Jaw of, 243, 259
— of evaporation, 263
— of fusion, 262
Thermodynamics of radiation, 293 et
seq.
Thermodynamics of refrigeration, 235
Thermodynamics of the atmosphere, 316
Thermodynamics, scope of, 237
— , second law of, 247
Thermodynamics, second law of,
— ? Clausius J s enunciation of, 248
, Kelvin's enunciation of, 248
— , preliminary statement of, 243
— , scope of, 247
— statement of, in terms of entropy,
255
Thermoelectric thermometry, 18 et seq.
Thermometer, alcohol, 2, 23
— , Beckmatm, 3
— , Callendar's compensated air, 6
— , chemical, 3
— , clinical, 3 ^
— , constant volume hydrogen, 7
— , gas, 4 et seq.
— , liquid, 13
— , maximum and minimum, 3
— , mercury', 1
—j petroleum ether, 23
— , platinum, 13 et seq,
—, secondary, 10, 27
— , standard gas, 7
— , standardisation of, 10
— , thermoelectric, 18 et seq.
— , vapour pressure, 22, 23
_, weight, 167
Thertnometric conductivity, 182
Thcrrnometr] , Char*. I
— , high temperature, 306 el seq.
— , low temperature, 22
— , resistance, 13 et seq.
— , thermoelectric, 18 *'' seq.
Thermopile, 279
Thcrmoregulator, 174
Thermostat, 174
Throttle valve, 210
Toluene thermostat, 174
Total heat of steam, 221
Total radiation pyrometers, 308
Transformation, adiabatic, 4%
Transport phenomena, 82
Triple point, 275
— for water, 275
Tropopausc, 317
— height of, 317
Troposphere, 317
— , pressure distribution in, 318
— ■ temperature distribution in, 319
t'urliin
U
i :gy, 256
i: round temperature, 188
tjaii of heat,
linn, production of high, 149
• lr:r Walls' equation of state, 91
. critical constants from, 95
— , deduction of, 91
— , defects in, 96
— , discussion of, 9S
— , JouleThomson effect from, 140,
268
— , methods of finding a and b, 92
Vaporisation, 104, 111 et seq,
— , latent heat of. 117 see also Latent
heat of vaporisation
Vapour compression machine, 127
— , efficiency of, 235
Vapour density, 122 et seq.
— of saturated vapour, 123
— , Victor Meyer's method, 123
Vapour pressure curve for water, 102,
321
— , discussion of results, 116
— , laws of, for mixture of liquids, 116
Vapour pressure measurements, dyna
mic or boiling point method, 114
— , static method, 113
Vapour pressure of water, 112
— over carved surfaces, 117
Vapour pressure thermometers, 22, 24
Velocity, average, 78
— , law of distribution of, 76
t probable
square, 74
ity of sound, 57 et seq.
Vibrational motion of diatomic mole
cules, 80
Viscosity, 83
— , discussion of the result, 84
W
Water, boiling point of, 11
— equivalent, 28
— , expansion of, 171
— , freezing point of, on absolute scale,
10
— , specific heat of, 40
— vapour in the atmosphere, 319
, method uf condensing, 324
— , vapour pressure of, 320
Watt's doubleacting engine, 208
— experimental condenser, 208
— governor, 210
— inventions, 207 et seq.
Weight thermometer, 167
Wiedemann and Franz's law, 190
W ion's constant, 304
Wien r s displacement law, 304
— , experimental verification of, 306
Work, graphical representation of, 213
— obtained in isothermal and adiabatic
expansion, 245
Working substance, 214
Xrays, 285
Zero, absolute, 10, 250